%   ORIGINAL: 'h4/thm/list/BIGUNION_IMAGE_set_SUBSET_'
% 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: !s ls f. 'h4/const/pred_set/SUBSET' ('h4/const/pred_set/BIGUNION' ('h4/const/pred_set/IMAGE' f ('h4/const/list/LIST_TO_SET' ls))) s <=> (!x. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ls) ==> 'h4/const/pred_set/SUBSET' (f 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: !s ls f. 'h4/const/pred_set/SUBSET' ('h4/const/pred_set/BIGUNION' ('h4/const/pred_set/IMAGE' f ('h4/const/list/LIST_TO_SET' ls))) s <=> (!x. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ls) ==> 'h4/const/pred_set/SUBSET' (happ f 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_3f11969,V_3f11965]: ![F, G]: (![X]: s(V_3f11965,happ(s(fun(V_3f11969,V_3f11965),F),s(V_3f11969,X))) = s(V_3f11965,happ(s(fun(V_3f11969,V_3f11965),G),s(V_3f11969,X))) => s(fun(V_3f11969,V_3f11965),F) = s(fun(V_3f11969,V_3f11965),G))).
fof('h4/thm/list/BIGUNION_IMAGE_set_SUBSET_', conjecture, ![B,A]: ![S, Ls, F]: (p(s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),'h4/const/pred_set/BIGUNION'(s(fun(fun(A,bool),bool),'h4/const/pred_set/IMAGE'(s(fun(B,fun(A,bool)),F),s(fun(B,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(B),Ls))))))),s(fun(A,bool),S)))) <=> ![X]: (p(s(bool,'h4/const/bool/IN'(s(B,X),s(fun(B,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(B),Ls)))))) => p(s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),happ(s(fun(B,fun(A,bool)),F),s(B,X))),s(fun(A,bool),S))))))).
