Theorem elghomlem1 10382

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
elghomlem1	|- ((G e. Grp /\ H e. Grp) -> (G GrpHom H) = S)

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

Proof of Theorem elghomlem1

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	fabexg 3653	. . 3 |- ((ran G e. V /\ ran H e. V) -> S e. V)
3		rnexg 3359	. . 3 |- (G e. Grp -> ran G e. V)
4		rnexg 3359	. . 3 |- (H e. Grp -> ran H e. V)
5	2, 3, 4	syl2an 454	. 2 |- ((G e. Grp /\ H e. Grp) -> S e. V)
6		rneq 3339	. . . . . 6 |- (g = G -> ran g = ran G)
7		feq2 3621	. . . . . 6 |- (ran g = ran G -> (f:ran g-->ran h <-> f:ran G-->ran h))
8	6, 7	syl 10	. . . . 5 |- (g = G -> (f:ran g-->ran h <-> f:ran G-->ran h))
9		opreq 3967	. . . . . . . . 9 |- (g = G -> (xgy) = (xGy))
10	9	fveq2d 3728	. . . . . . . 8 |- (g = G -> (f` (xgy)) = (f` (xGy)))
11	10	eqeq2d 1486	. . . . . . 7 |- (g = G -> (((f` x)h(f` y)) = (f` (xgy)) <-> ((f` x)h(f` y)) = (f` (xGy))))
12	6, 11	raleq12d 1794	. . . . . 6 |- (g = G -> (A.y e. ran g((f` x)h(f` y)) = (f` (xgy)) <-> A.y e. ran G((f` x)h(f` y)) = (f` (xGy))))
13	6, 12	raleq12d 1794	. . . . 5 |- (g = G -> (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))))
14	8, 13	anbi12d 628	. . . 4 |- (g = G -> ((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)))))
15	14	abbidv 1577	. . 3 |- (g = G -> {f | (f:ran g-->ran h /\ A.x e. ran gA.y e. ran g((f` x)h(f` y)) = (f` (xgy)))} = {f | (f:ran G-->ran h /\ A.x e. ran GA.y e. ran G((f` x)h(f` y)) = (f` (xGy)))})
16		rneq 3339	. . . . . . 7 |- (h = H -> ran h = ran H)
17		feq3 3622	. . . . . . 7 |- (ran h = ran H -> (f:ran G-->ran h <-> f:ran G-->ran H))
18	16, 17	syl 10	. . . . . 6 |- (h = H -> (f:ran G-->ran h <-> f:ran G-->ran H))
19		opreq 3967	. . . . . . . 8 |- (h = H -> ((f` x)h(f` y)) = ((f` x)H(f` y)))
20	19	eqeq1d 1483	. . . . . . 7 |- (h = H -> (((f` x)h(f` y)) = (f` (xGy)) <-> ((f` x)H(f` y)) = (f` (xGy))))
21	20	2ralbidv 1680	. . . . . 6 |- (h = H -> (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))))
22	18, 21	anbi12d 628	. . . . 5 |- (h = H -> ((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)))))
23	22	abbidv 1577	. . . 4 |- (h = H -> {f | (f:ran G-->ran h /\ A.x e. ran GA.y e. ran G((f` x)h(f` y)) = (f` (xGy)))} = {f | (f:ran G-->ran H /\ A.x e. ran GA.y e. ran G((f` x)H(f` y)) = (f` (xGy)))})
24	23, 1	syl6eqr 1525	. . 3 |- (h = H -> {f | (f:ran G-->ran h /\ A.x e. ran GA.y e. ran G((f` x)h(f` y)) = (f` (xGy)))} = S)
25		df-ghom 10380	. . 3 |- GrpHom = {<.<.g, h>., z>. | ((g e. Grp /\ h e. Grp) /\ z = {f | (f:ran g-->ran h /\ A.x e. ran gA.y e. ran g((f` x)h(f` y)) = (f` (xgy)))})}
26	15, 24, 25	oprabval2g 4027	. 2 |- ((G e. Grp /\ H e. Grp /\ S e. V) -> (G GrpHom H) = S)
27	5, 26	mpd3an3 917	1 |- ((G e. Grp /\ H e. Grp) -> (G GrpHom H) = S)

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:  elghomlem2 10383

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-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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