%   ORIGINAL: 'h4/thm/pred_set/BIJ_IFF_INV_'
% 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 s t. 'h4/const/pred_set/BIJ' f s t <=> (!x. 'h4/const/bool/IN' x s ==> 'h4/const/bool/IN' (f x) t) /\ (?g. (!x. 'h4/const/bool/IN' x t ==> 'h4/const/bool/IN' (g x) s) /\ (!x. 'h4/const/bool/IN' x s ==> g (f x) = x) /\ (!x. 'h4/const/bool/IN' x t ==> f (g x) = x))
%   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: !f s t. 'h4/const/pred_set/BIJ' f s t <=> (!x. 'h4/const/bool/IN' x s ==> 'h4/const/bool/IN' (happ f x) t) /\ (?g. (!x. 'h4/const/bool/IN' x t ==> 'h4/const/bool/IN' (happ g x) s) /\ (!x. 'h4/const/bool/IN' x s ==> happ g (happ f x) = x) /\ (!x. 'h4/const/bool/IN' x t ==> happ f (happ g x) = x))
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_3f18970,V_3f18966]: ![F, G]: (![X]: s(V_3f18966,happ(s(fun(V_3f18970,V_3f18966),F),s(V_3f18970,X))) = s(V_3f18966,happ(s(fun(V_3f18970,V_3f18966),G),s(V_3f18970,X))) => s(fun(V_3f18970,V_3f18966),F) = s(fun(V_3f18970,V_3f18966),G))).
fof('h4/thm/pred_set/BIJ_IFF_INV_', conjecture, ![A,B]: ![F, S, T]: (p(s(bool,'h4/const/pred_set/BIJ'(s(fun(A,B),F),s(fun(A,bool),S),s(fun(B,bool),T)))) <=> (![X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) => p(s(bool,'h4/const/bool/IN'(s(B,happ(s(fun(A,B),F),s(A,X))),s(fun(B,bool),T))))) & ?[G]: (![X]: (p(s(bool,'h4/const/bool/IN'(s(B,X),s(fun(B,bool),T)))) => p(s(bool,'h4/const/bool/IN'(s(A,happ(s(fun(B,A),G),s(B,X))),s(fun(A,bool),S))))) & (![X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) => s(A,happ(s(fun(B,A),G),s(B,happ(s(fun(A,B),F),s(A,X))))) = s(A,X)) & ![X]: (p(s(bool,'h4/const/bool/IN'(s(B,X),s(fun(B,bool),T)))) => s(B,happ(s(fun(A,B),F),s(A,happ(s(fun(B,A),G),s(B,X))))) = s(B,X))))))).
