%   ORIGINAL: h4/tc/FDOM__RDOM
% 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
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/finite__map/FUN__FMAP__DEF: !f P. h4/pred__set/FINITE P ==> h4/finite__map/FDOM (h4/finite__map/FUN__FMAP f P) = P /\ (!x. h4/bool/IN x P ==> h4/finite__map/FAPPLY (h4/finite__map/FUN__FMAP f P) x = f x)
% Assm: h4/tc/RELN__TO__FMAP0: !R. h4/tc/RELN__TO__FMAP R = h4/finite__map/FUN__FMAP R (h4/relation/RDOM R)
% Goal: !R. h4/pred__set/FINITE (h4/relation/RDOM R) ==> h4/finite__map/FDOM (h4/tc/RELN__TO__FMAP R) = h4/relation/RDOM R
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_finiteu_u_maps_FUNu_u_FMAPu_u_DEF]: !f P. h4/pred__set/FINITE P ==> h4/finite__map/FDOM (h4/finite__map/FUN__FMAP f P) = P /\ (!x. h4/bool/IN x P ==> h4/finite__map/FAPPLY (h4/finite__map/FUN__FMAP f P) x = happ f x)
% Assm [h4s_tcs_RELNu_u_TOu_u_FMAP0]: !R. h4/tc/RELN__TO__FMAP R = h4/finite__map/FUN__FMAP R (h4/relation/RDOM R)
% Goal: !R. h4/pred__set/FINITE (h4/relation/RDOM R) ==> h4/finite__map/FDOM (h4/tc/RELN__TO__FMAP R) = h4/relation/RDOM R
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q189648,TV_Q189644]: ![V_f, V_g]: (![V_x]: s(TV_Q189644,happ(s(t_fun(TV_Q189648,TV_Q189644),V_f),s(TV_Q189648,V_x))) = s(TV_Q189644,happ(s(t_fun(TV_Q189648,TV_Q189644),V_g),s(TV_Q189648,V_x))) => s(t_fun(TV_Q189648,TV_Q189644),V_f) = s(t_fun(TV_Q189648,TV_Q189644),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_finiteu_u_maps_FUNu_u_FMAPu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) => (s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_P))))) = s(t_fun(TV_u_27a,t_bool),V_P) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_tcs_RELNu_u_TOu_u_FMAP0, axiom, ![TV_u_27a]: ![V_R]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_tcs_relnu_u_tou_u_fmap(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ch4s_tcs_FDOMu_u_RDOM, conjecture, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))) => s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_tcs_relnu_u_tou_u_fmap(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))).
