%   ORIGINAL: 'h4/thm/pred_set/IN_IMAGE_'
% 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: !y s f. 'h4/const/bool/IN' y ('h4/const/pred_set/IMAGE' f s) <=> (?x. y = f x /\ 'h4/const/bool/IN' x 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: !y s f. 'h4/const/bool/IN' y ('h4/const/pred_set/IMAGE' f s) <=> (?x. y = happ f x /\ 'h4/const/bool/IN' x 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_3f18070,V_3f18066]: ![F, G]: (![X]: s(V_3f18066,happ(s(fun(V_3f18070,V_3f18066),F),s(V_3f18070,X))) = s(V_3f18066,happ(s(fun(V_3f18070,V_3f18066),G),s(V_3f18070,X))) => s(fun(V_3f18070,V_3f18066),F) = s(fun(V_3f18070,V_3f18066),G))).
fof('h4/thm/pred_set/IN_IMAGE_', conjecture, ![B,A]: ![Y, S, F]: (p(s(bool,'h4/const/bool/IN'(s(B,Y),s(fun(B,bool),'h4/const/pred_set/IMAGE'(s(fun(A,B),F),s(fun(A,bool),S)))))) <=> ?[X]: (s(B,Y) = s(B,happ(s(fun(A,B),F),s(A,X))) & p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S))))))).
