Theorem elghomlem2 10194

Description: Lemma for elghom 10195.

Hypothesis

Ref	Expression
elghomlem1.1	|- S = {f | (f:ran G-->ran H /\ A.x e. ran GA.y e. ran G((f` x)H(f` y)) = (f` (xGy)))}

Assertion

Ref	Expression
elghomlem2	|- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))

Distinct variable groups:   f,F,x,y   f,G,x,y   f,H,x,y

Proof of Theorem elghomlem2

Step	Hyp	Ref	Expression
1		elghomlem1.1	. . . 4 |- S = {f | (f:ran G-->ran H /\ A.x e. ran GA.y e. ran G((f` x)H(f` y)) = (f` (xGy)))}
2	1	elghomlem1 10193	. . 3 |- ((G e. Grp /\ H e. Grp) -> (G GrpHom H) = S)
3	2	eleq2d 1964	. 2 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> F e. S))
4		elisset 2299	. . . . 5 |- (F e. S -> F e. _V)
5		feq1 4551	. . . . . . . 8 |- (f = F -> (f:ran G-->ran H <-> F:ran G-->ran H))
6		fveq1 4680	. . . . . . . . . . 11 |- (f = F -> (f` x) = (F` x))
7		fveq1 4680	. . . . . . . . . . 11 |- (f = F -> (f` y) = (F` y))
8	6, 7	opreq12d 4900	. . . . . . . . . 10 |- (f = F -> ((f` x)H(f` y)) = ((F` x)H(F` y)))
9		fveq1 4680	. . . . . . . . . 10 |- (f = F -> (f` (xGy)) = (F` (xGy)))
10	8, 9	eqeq12d 1899	. . . . . . . . 9 |- (f = F -> (((f` x)H(f` y)) = (f` (xGy)) <-> ((F` x)H(F` y)) = (F` (xGy))))
11	10	2ralbidv 2140	. . . . . . . 8 |- (f = F -> (A.x e. ran GA.y e. ran G((f` x)H(f` y)) = (f` (xGy)) <-> A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))))
12	5, 11	anbi12d 690	. . . . . . 7 |- (f = F -> ((f:ran G-->ran H /\ A.x e. ran GA.y e. ran G((f` x)H(f` y)) = (f` (xGy))) <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
13	12, 1	elab2g 2406	. . . . . 6 |- (F e. _V -> (F e. S <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
14	13	biimpd 170	. . . . 5 |- (F e. _V -> (F e. S -> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
15	4, 14	mpcom 60	. . . 4 |- (F e. S -> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))))
16		rnexg 4207	. . . . . . 7 |- (G e. Grp -> ran G e. _V)
17		fex 4595	. . . . . . . 8 |- ((F:ran G-->ran H /\ ran G e. _V) -> F e. _V)
18	17	expcom 403	. . . . . . 7 |- (ran G e. _V -> (F:ran G-->ran H -> F e. _V))
19	16, 18	syl 12	. . . . . 6 |- (G e. Grp -> (F:ran G-->ran H -> F e. _V))
20	19	adantrd 427	. . . . 5 |- (G e. Grp -> ((F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))) -> F e. _V))
21	13	biimprd 171	. . . . 5 |- (F e. _V -> ((F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))) -> F e. S))
22	20, 21	syli 65	. . . 4 |- (G e. Grp -> ((F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))) -> F e. S))
23	15, 22	impbid2 576	. . 3 |- (G e. Grp -> (F e. S <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
24	23	adantr 425	. 2 |- ((G e. Grp /\ H e. Grp) -> (F e. S <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
25	3, 24	bitrd 587	1 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  Grpcgr 9311   GrpHom cghom 10189

This theorem is referenced by:  elghom 10195

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-ghom 10190

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