%   ORIGINAL: 'h4/thm/pred_set/FINITELY_INJECTIVE_IMAGE_FINITE_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !f. (!x. 'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (\y. 'h4/const/pair/,' y (x = f y)))) ==> (!s. 'h4/const/pred_set/FINITE' ('h4/const/pred_set/IMAGE' f s) <=> 'h4/const/pred_set/FINITE' s)
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!x f y. ?v. (v <=> x = happ f y) /\ happ (happ (happ _0 x) f) y = 'h4/const/pair/,' y v) ==> (!f. (!x. 'h4/const/pred_set/FINITE' ('h4/const/pred_set/GSPEC' (happ (happ _0 x) f))) ==> (!s. 'h4/const/pred_set/FINITE' ('h4/const/pred_set/IMAGE' f s) <=> 'h4/const/pred_set/FINITE' s))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f19395,V_3f19391]: ![F, G]: (![X]: s(V_3f19391,happ(s(fun(V_3f19395,V_3f19391),F),s(V_3f19395,X))) = s(V_3f19391,happ(s(fun(V_3f19395,V_3f19391),G),s(V_3f19395,X))) => s(fun(V_3f19395,V_3f19391),F) = s(fun(V_3f19395,V_3f19391),G))).
fof('h4/thm/pred_set/FINITELY_INJECTIVE_IMAGE_FINITE_', conjecture, ![B,A]: ![V__0]: (![X, F, Y]: ?[V]: ((p(s(bool,V)) <=> s(B,X) = s(B,happ(s(fun(A,B),F),s(A,Y)))) & s('h4/type/pair/prod'(A,bool),happ(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,B),fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(B,fun(fun(A,B),fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(B,X))),s(fun(A,B),F))),s(A,Y))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,Y),s(bool,V)))) => ![F]: (![X]: p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,B),fun(A,'h4/type/pair/prod'(A,bool))),happ(s(fun(B,fun(fun(A,B),fun(A,'h4/type/pair/prod'(A,bool)))),V__0),s(B,X))),s(fun(A,B),F)))))))) => ![S]: s(bool,'h4/const/pred_set/FINITE'(s(fun(B,bool),'h4/const/pred_set/IMAGE'(s(fun(A,B),F),s(fun(A,bool),S))))) = s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))))).
