%   ORIGINAL: 'h4/thm/pred_set/IMAGE_11_'
% 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: !s2 s1 f. (!x y. f x = f y <=> x = y) ==> ('h4/const/pred_set/IMAGE' f s1 = 'h4/const/pred_set/IMAGE' f s2 <=> s1 = s2)
%   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: !s2 s1 f. (!x y. happ f x = happ f y <=> x = y) ==> ('h4/const/pred_set/IMAGE' f s1 = 'h4/const/pred_set/IMAGE' f s2 <=> s1 = s2)
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_3f18370,V_3f18366]: ![F, G]: (![X]: s(V_3f18366,happ(s(fun(V_3f18370,V_3f18366),F),s(V_3f18370,X))) = s(V_3f18366,happ(s(fun(V_3f18370,V_3f18366),G),s(V_3f18370,X))) => s(fun(V_3f18370,V_3f18366),F) = s(fun(V_3f18370,V_3f18366),G))).
fof('h4/thm/pred_set/IMAGE_11_', conjecture, ![B,A]: ![S2, S1, F]: (![X, Y]: (s(B,happ(s(fun(A,B),F),s(A,X))) = s(B,happ(s(fun(A,B),F),s(A,Y))) <=> s(A,X) = s(A,Y)) => (s(fun(B,bool),'h4/const/pred_set/IMAGE'(s(fun(A,B),F),s(fun(A,bool),S1))) = s(fun(B,bool),'h4/const/pred_set/IMAGE'(s(fun(A,B),F),s(fun(A,bool),S2))) <=> s(fun(A,bool),S1) = s(fun(A,bool),S2)))).
