:: FUNCT_1 semantic presentation begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is :::"Function-like"::: means :: FUNCT_1:def 1 (Bool "for" (Set (Var "x")) "," (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set (Var "y1")) ($#r1_hidden :::"="::: ) (Set (Var "y2")))); end; :: deftheorem defines :::"Function-like"::: FUNCT_1:def 1 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_funct_1 :::"Function-like"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "y1")) ($#r1_hidden :::"="::: ) (Set (Var "y2")))) ")" )); registration cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionmode Function is ($#v1_funct_1 :::"Function-like"::: ) ($#m1_hidden :::"Relation":::); end; registrationlet "a", "b" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "a" "," "b" ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) -> ($#v1_funct_1 :::"Function-like"::: ) ; end; scheme :: FUNCT_1:sch 1 GraphFunc{ F1() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool P1[(Set (Var "x")) "," (Set (Var "y"))]) ")" ) ")" ))) provided (Bool "for" (Set (Var "x")) "," (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool P1[(Set (Var "x")) "," (Set (Var "y1"))]) & (Bool P1[(Set (Var "x")) "," (Set (Var "y2"))])) "holds" (Bool (Set (Var "y1")) ($#r1_hidden :::"="::: ) (Set (Var "y2")))) proof end; definitionlet "f" be ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; func "f" :::"."::: "x" -> ($#m1_hidden :::"set"::: ) means :: FUNCT_1:def 2 (Bool (Set ($#k4_tarski :::"["::: ) "x" "," it ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "f") if (Bool "x" ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) otherwise (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"."::: FUNCT_1:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "implies" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) "iff" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "b3")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "f"))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))))) "implies" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) "iff" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) ")" ")" ))); theorem :: FUNCT_1:1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ))) ; theorem :: FUNCT_1:2 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); redefine func :::"rng"::: "f" means :: FUNCT_1:def 3 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" )); end; :: deftheorem defines :::"rng"::: FUNCT_1:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" )) ")" ))); theorem :: FUNCT_1:3 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; theorem :: FUNCT_1:4 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_tarski :::"}"::: ) )))) ; scheme :: FUNCT_1:sch 2 FuncEx{ F1() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))]) ")" ) ")" )) provided (Bool "for" (Set (Var "x")) "," (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool P1[(Set (Var "x")) "," (Set (Var "y1"))]) & (Bool P1[(Set (Var "x")) "," (Set (Var "y2"))])) "holds" (Bool (Set (Var "y1")) ($#r1_hidden :::"="::: ) (Set (Var "y2")))) and (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool P1[(Set (Var "x")) "," (Set (Var "y"))]))) proof end; scheme :: FUNCT_1:sch 3 Lambda{ F1() -> ($#m1_hidden :::"set"::: ) , F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set F2 "(" (Set (Var "x")) ")" )) ")" ) ")" )) proof end; theorem :: FUNCT_1:5 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ")" )))) ; theorem :: FUNCT_1:6 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FUNCT_1:7 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: FUNCT_1:8 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" (Bool (Set (Var "Y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) ")" ))) ; theorem :: FUNCT_1:9 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" )) "holds" (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; notationlet "f", "g" be ($#m1_hidden :::"Function":::); synonym "g" :::"*"::: "f" for "f" :::"*"::: "g"; end; registrationlet "f", "g" be ($#m1_hidden :::"Function":::); cluster (Set "g" ($#k3_relat_1 :::"*"::: ) "f") -> ($#v1_funct_1 :::"Function-like"::: ) ; end; theorem :: FUNCT_1:10 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) ")" ) ")" ) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" )) "holds" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))))) ; theorem :: FUNCT_1:11 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) ")" ) ")" ))) ; theorem :: FUNCT_1:12 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))))) ; theorem :: FUNCT_1:13 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))))) ; theorem :: FUNCT_1:14 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")))))) ; theorem :: FUNCT_1:15 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) ; theorem :: FUNCT_1:16 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool "(" "for" (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "Y"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Var "Y"))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "h")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "h"))) ")" )) "holds" (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_relat_1 :::"id"::: ) "X") -> ($#v1_funct_1 :::"Function-like"::: ) ; end; theorem :: FUNCT_1:17 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ) ")" ))) ; theorem :: FUNCT_1:18 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ; theorem :: FUNCT_1:19 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:20 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: FUNCT_1:21 (Bool "for" (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "Y")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" ) ")" ))) ; theorem :: FUNCT_1:22 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )))) ; theorem :: FUNCT_1:23 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" )))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); attr "f" is :::"one-to-one"::: means :: FUNCT_1:def 4 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x2"))))) "holds" (Bool (Set (Var "x1")) ($#r1_hidden :::"="::: ) (Set (Var "x2")))); end; :: deftheorem defines :::"one-to-one"::: FUNCT_1:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x2"))))) "holds" (Bool (Set (Var "x1")) ($#r1_hidden :::"="::: ) (Set (Var "x2")))) ")" )); theorem :: FUNCT_1:24 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FUNCT_1:25 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FUNCT_1:26 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FUNCT_1:27 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) ")" )) ; theorem :: FUNCT_1:28 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:29 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")))) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_relat_1 :::"id"::: ) "X") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: FUNCT_1:30 canceled; theorem :: FUNCT_1:31 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ))))) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; registration cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v2_funct_1 :::"one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"Function":::); cluster (Set "f" ($#k2_relat_1 :::"~"::: ) ) -> ($#v1_funct_1 :::"Function-like"::: ) ; end; definitionlet "f" be ($#m1_hidden :::"Function":::); assume (Bool (Set (Const "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ; func "f" :::"""::: -> ($#m1_hidden :::"Function":::) equals :: FUNCT_1:def 5 (Set "f" ($#k2_relat_1 :::"~"::: ) ); end; :: deftheorem defines :::"""::: FUNCT_1:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_relat_1 :::"~"::: ) ))); theorem :: FUNCT_1:32 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ) ")" ) ")" ) ")" ))) ; theorem :: FUNCT_1:33 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ))) ")" )) ; theorem :: FUNCT_1:34 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ))) ; theorem :: FUNCT_1:35 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y")) ")" ))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) ")" ))) ; theorem :: FUNCT_1:36 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) ")" )) ; theorem :: FUNCT_1:37 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) ")" )) ; theorem :: FUNCT_1:38 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))) "iff" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" )) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ))) ; theorem :: FUNCT_1:39 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ))) ")" )) ; theorem :: FUNCT_1:40 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; registrationlet "f" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"Function":::); cluster (Set "f" ($#k2_funct_1 :::"""::: ) ) -> ($#v2_funct_1 :::"one-to-one"::: ) ; let "g" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"Function":::); cluster (Set "g" ($#k3_relat_1 :::"*"::: ) "f") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: FUNCT_1:41 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ))) ; theorem :: FUNCT_1:42 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ))) ; theorem :: FUNCT_1:43 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ; theorem :: FUNCT_1:44 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k2_funct_1 :::"""::: ) ")" )))) ; theorem :: FUNCT_1:45 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "X"))))) ; registrationlet "f" be ($#m1_hidden :::"Function":::); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_funct_1 :::"Function-like"::: ) ; end; theorem :: FUNCT_1:46 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:47 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: FUNCT_1:48 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: FUNCT_1:49 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: FUNCT_1:50 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ))))) ; theorem :: FUNCT_1:51 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "Y")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")))) ")" ))) ; theorem :: FUNCT_1:52 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; registrationlet "Y" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"Function":::); cluster (Set "Y" ($#k6_relat_1 :::"|`"::: ) "f") -> ($#v1_funct_1 :::"Function-like"::: ) ; end; theorem :: FUNCT_1:53 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" ) ")" ) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ) ")" ))) ; theorem :: FUNCT_1:54 (Bool "for" (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" ))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" ) ")" ))) ; theorem :: FUNCT_1:55 (Bool "for" (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: FUNCT_1:56 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: FUNCT_1:57 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool "(" (Bool (Set (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set "(" (Set (Var "X")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "X")) ($#k6_relat_1 :::"|`"::: ) (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")))) ")" ))) ; theorem :: FUNCT_1:58 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); let "X" be ($#m1_hidden :::"set"::: ) ; redefine func "f" :::".:"::: "X" means :: FUNCT_1:def 6 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" )); end; :: deftheorem defines :::".:"::: FUNCT_1:def 6 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")))) "iff" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" )) ")" )))); theorem :: FUNCT_1:59 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k9_relat_1 :::"Im"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FUNCT_1:60 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "x1")) "," (Set (Var "x2")) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x1")) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x2")) ")" ) ($#k2_tarski :::"}"::: ) )))) ; theorem :: FUNCT_1:61 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:62 (Bool "for" (Set (Var "X1")) "," (Set (Var "X2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "X1")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X2")) ")" ))))) ; theorem :: FUNCT_1:63 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "(" "for" (Set (Var "X1")) "," (Set (Var "X2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "X1")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X2")) ")" ))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FUNCT_1:64 (Bool "for" (Set (Var "X1")) "," (Set (Var "X2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "X1")) ($#k6_subset_1 :::"\"::: ) (Set (Var "X2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X1")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X2")) ")" ))))) ; theorem :: FUNCT_1:65 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "(" "for" (Set (Var "X1")) "," (Set (Var "X2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "X1")) ($#k6_subset_1 :::"\"::: ) (Set (Var "X2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X1")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X2")) ")" ))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FUNCT_1:66 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "Y"))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "Y")))))) ; theorem :: FUNCT_1:67 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" (Set (Var "Y")) ($#k6_relat_1 :::"|`"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "Y")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ))))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); let "Y" be ($#m1_hidden :::"set"::: ) ; redefine func "f" :::"""::: "Y" means :: FUNCT_1:def 7 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) "Y") ")" ) ")" )); end; :: deftheorem defines :::"""::: FUNCT_1:def 7 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" ) ")" )) ")" )))); theorem :: FUNCT_1:68 (Bool "for" (Set (Var "Y1")) "," (Set (Var "Y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "Y1")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y2")) ")" ))))) ; theorem :: FUNCT_1:69 (Bool "for" (Set (Var "Y1")) "," (Set (Var "Y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "Y1")) ($#k6_subset_1 :::"\"::: ) (Set (Var "Y2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y1")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y2")) ")" ))))) ; theorem :: FUNCT_1:70 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool (Set (Set "(" (Set (Var "R")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" ))))) ; theorem :: FUNCT_1:71 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "A"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "B")))))) ; theorem :: FUNCT_1:72 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "R")))) "iff" (Bool (Set (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; theorem :: FUNCT_1:73 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) "st" (Bool (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) "holds" (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "R")))))) ; theorem :: FUNCT_1:74 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ")" ) "iff" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FUNCT_1:75 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))))) ; theorem :: FUNCT_1:76 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "R"))))) "holds" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "R")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ))))) ; theorem :: FUNCT_1:77 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Y"))))) ; theorem :: FUNCT_1:78 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "Y")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ")" ))))) ; theorem :: FUNCT_1:79 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))))) ; theorem :: FUNCT_1:80 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))))) ; theorem :: FUNCT_1:81 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set "(" (Set (Var "R")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" ))))) ; theorem :: FUNCT_1:82 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "X"))))) ; theorem :: FUNCT_1:83 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FUNCT_1:84 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:85 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "Y")))))) ; theorem :: FUNCT_1:86 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "Y"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Var "Y"))) & (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "h")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "h"))))) ; theorem :: FUNCT_1:87 (Bool "for" (Set (Var "X1")) "," (Set (Var "X2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X2")))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X2"))))) ; theorem :: FUNCT_1:88 (Bool "for" (Set (Var "Y1")) "," (Set (Var "Y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y2")))) & (Bool (Set (Var "Y1")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "Y1")) ($#r1_tarski :::"c="::: ) (Set (Var "Y2"))))) ; theorem :: FUNCT_1:89 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )))) ")" )) ; theorem :: FUNCT_1:90 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#m1_hidden :::"Relation":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "R"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set (Var "R")) ($#k3_relat_1 :::"*"::: ) (Set (Var "S")) ")" ) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "S")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ))))) ; theorem :: FUNCT_1:91 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")))) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "Y"))))) ; begin theorem :: FUNCT_1:92 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); redefine attr "f" is :::"empty-yielding"::: means :: FUNCT_1:def 8 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) "is" ($#v1_xboole_0 :::"empty"::: ) )); end; :: deftheorem defines :::"empty-yielding"::: FUNCT_1:def 8 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_relat_1 :::"empty-yielding"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) ")" )); definitionlet "F" be ($#m1_hidden :::"Function":::); redefine attr "F" is :::"non-empty"::: means :: FUNCT_1:def 9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "F"))) "holds" (Bool "not" (Bool (Set "F" ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))); end; :: deftheorem defines :::"non-empty"::: FUNCT_1:def 9 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_relat_1 :::"non-empty"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool "not" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v2_relat_1 :::"non-empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; scheme :: FUNCT_1:sch 4 LambdaB{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set F2 "(" (Set (Var "d")) ")" )) ")" ) ")" )) proof end; registrationlet "f" be ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::); cluster (Set ($#k10_xtuple_0 :::"rng"::: ) "f") -> ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ; end; definitionlet "f" be ($#m1_hidden :::"Function":::); attr "f" is :::"constant"::: means :: FUNCT_1:def 10 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "y"))))); end; :: deftheorem defines :::"constant"::: FUNCT_1:def 10 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_1 :::"constant"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))))) ")" )); theorem :: FUNCT_1:93 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "B")))))) ; theorem :: FUNCT_1:94 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "X"))))) ; definitionlet "f", "g" be ($#m1_hidden :::"Function":::); redefine pred "f" :::"="::: "g" means :: FUNCT_1:def 11 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) "f") ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "g")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set "g" ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ); end; :: deftheorem defines :::"="::: FUNCT_1:def 11 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ) ")" )); registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v2_relat_1 :::"non-empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "a" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Const "a"))); cluster (Set "a" ($#k1_funct_1 :::"."::: ) "i") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "f" be ($#m1_hidden :::"Function":::); cluster -> ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "f"); end; theorem :: FUNCT_1:95 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k5_relat_1 :::"|"::: ) (Set (Var "D")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))))) ")" ))) ; theorem :: FUNCT_1:96 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:97 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "X")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FUNCT_1:98 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "X")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )))) ; registration cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v3_funct_1 :::"constant"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f" be ($#v3_funct_1 :::"constant"::: ) ($#m1_hidden :::"Function":::); cluster (Set ($#k10_xtuple_0 :::"rng"::: ) "f") -> ($#v1_zfmisc_1 :::"trivial"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#m1_hidden :::"Function":::); cluster (Set ($#k10_xtuple_0 :::"rng"::: ) "f") -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_zfmisc_1 :::"trivial"::: ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v3_funct_1 :::"constant"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FUNCT_1:99 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set (Var "F")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "F")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_relat_1 :::"*"::: ) (Set (Var "F")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:100 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X1")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "F")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_relat_1 :::"*"::: ) (Set (Var "F")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set (Var "X1")) ")" ) ")" ))))) ; theorem :: FUNCT_1:101 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "G")) ($#k3_relat_1 :::"*"::: ) (Set (Var "F")) ")" ))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Set (Var "F")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "G")))) ")" ) ")" ))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); assume (Bool "(" (Bool (Bool "not" (Set (Const "f")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Const "f")) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" ) ; func :::"the_value_of"::: "f" -> ($#m1_hidden :::"set"::: ) means :: FUNCT_1:def 12 (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool it ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )); end; :: deftheorem defines :::"the_value_of"::: FUNCT_1:def 12 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_funct_1 :::"the_value_of"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" ))); registrationlet "X", "Y" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "X" ($#v4_relat_1 :::"-defined"::: ) "Y" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FUNCT_1:102 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" (Set (Var "b1")) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; definitionlet "IT" be ($#m1_hidden :::"set"::: ) ; attr "IT" is :::"functional"::: means :: FUNCT_1:def 13 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "IT")) "holds" (Bool (Set (Var "x")) "is" ($#m1_hidden :::"Function":::))); end; :: deftheorem defines :::"functional"::: FUNCT_1:def 13 : (Bool "for" (Set (Var "IT")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v4_funct_1 :::"functional"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Var "x")) "is" ($#m1_hidden :::"Function":::))) ")" )); registration cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v4_funct_1 :::"functional"::: ) for ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"Function":::); cluster (Set ($#k1_tarski :::"{"::: ) "f" ($#k1_tarski :::"}"::: ) ) -> ($#v4_funct_1 :::"functional"::: ) ; let "g" be ($#m1_hidden :::"Function":::); cluster (Set ($#k2_tarski :::"{"::: ) "f" "," "g" ($#k2_tarski :::"}"::: ) ) -> ($#v4_funct_1 :::"functional"::: ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "P" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" "P"; end; registrationlet "A" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v4_funct_1 :::"functional"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "A"); end; definitionlet "g", "f" be ($#m1_hidden :::"Function":::); attr "f" is "g" :::"-compatible"::: means :: FUNCT_1:def 14 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set "g" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))))); end; :: deftheorem defines :::"-compatible"::: FUNCT_1:def 14 : (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" (Set (Var "g")) ($#v5_funct_1 :::"-compatible"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))))) ")" )); theorem :: FUNCT_1:103 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" (Set (Var "g")) ($#v5_funct_1 :::"-compatible"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "g")) "is" ($#v2_relat_1 :::"non-empty"::: ) )) ; theorem :: FUNCT_1:104 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" (Set (Var "f")) ($#v5_funct_1 :::"-compatible"::: ) )) ; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"Function":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) "f" ($#v5_funct_1 :::"-compatible"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"Function":::); let "g" be (Set (Const "f")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"Function":::); cluster (Set "g" ($#k5_relat_1 :::"|"::: ) "X") -> "f" ($#v5_funct_1 :::"-compatible"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v2_relat_1 :::"non-empty"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FUNCT_1:105 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "g")) "being" (Set (Var "b1")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))))) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "f" be (Set (Const "X")) ($#v4_relat_1 :::"-defined"::: ) ($#m1_hidden :::"Function":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) "f" ($#v5_funct_1 :::"-compatible"::: ) -> "X" ($#v4_relat_1 :::"-defined"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FUNCT_1:106 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" (Set (Var "b1")) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X"))))) ; theorem :: FUNCT_1:107 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; registrationlet "A" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_hidden :::"set"::: ) ; let "F" be (Set (Const "A")) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); cluster (Set "F" ($#k1_funct_1 :::"."::: ) "x") -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ; end; theorem :: FUNCT_1:108 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_1:109 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; registrationlet "f" be ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"Function":::); cluster (Set ($#k9_xtuple_0 :::"dom"::: ) "f") -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ; end; theorem :: FUNCT_1:110 (Bool "for" (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set (Var "B"))))) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) where g "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "B")) : (Bool verum) "}" )) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" ) where g "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "B")) : (Bool verum) "}" )) ")" ))) ; theorem :: FUNCT_1:111 (Bool "for" (Set (Var "M")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "M"))) & (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ) ")" ))) ; scheme :: FUNCT_1:sch 5 NonUniqBoundFuncEx{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))]) ")" ) ")" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" )) & (Bool P1[(Set (Var "x")) "," (Set (Var "y"))]) ")" ))) proof end; registrationlet "f" be ($#v3_relat_1 :::"empty-yielding"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k1_funct_1 :::"."::: ) "x") -> ($#v1_xboole_0 :::"empty"::: ) ; end;