Theorem elghomlem2 10383

Description: Lemma for elghom 10384.

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 10382	. . 3 |- ((G e. Grp /\ H e. Grp) -> (G GrpHom H) = S)
3	2	eleq2d 1541	. 2 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> F e. S))
4		elisset 1817	. . . . 5 |- (F e. S -> F e. V)
5		feq1 3620	. . . . . . . 8 |- (f = F -> (f:ran G-->ran H <-> F:ran G-->ran H))
6		fveq1 3723	. . . . . . . . . . 11 |- (f = F -> (f` x) = (F` x))
7		fveq1 3723	. . . . . . . . . . 11 |- (f = F -> (f` y) = (F` y))
8	6, 7	opreq12d 3978	. . . . . . . . . 10 |- (f = F -> ((f` x)H(f` y)) = ((F` x)H(F` y)))
9		fveq1 3723	. . . . . . . . . 10 |- (f = F -> (f` (xGy)) = (F` (xGy)))
10	8, 9	eqeq12d 1489	. . . . . . . . 9 |- (f = F -> (((f` x)H(f` y)) = (f` (xGy)) <-> ((F` x)H(F` y)) = (F` (xGy))))
11	10	2ralbidv 1680	. . . . . . . 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 628	. . . . . . 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 1900	. . . . . 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 153	. . . . 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 49	. . . 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 3359	. . . . . . 7 |- (G e. Grp -> ran G e. V)
17		fex 3652	. . . . . . . 8 |- ((F:ran G-->ran H /\ ran G e. V) -> F e. V)
18	17	expcom 374	. . . . . . 7 |- (ran G e. V -> (F:ran G-->ran H -> F e. V))
19	16, 18	syl 10	. . . . . 6 |- (G e. Grp -> (F:ran G-->ran H -> F e. V))
20	19	adantrd 391	. . . . 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 154	. . . . 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 54	. . . 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 518	. . 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 389	. 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 528	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 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  Vcvv 1811  ran crn 3171  -->wf 3178  ` cfv 3182  (class class class)co 3963  Grpcgr 8033   GrpHom cghom 10378

This theorem is referenced by:  elghom 10384

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-ghom 10380

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