Theorem elghomlem2 10378

Description: Lemma for elghom 10379.

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 10377	. . 3 |- ((G e. Grp /\ H e. Grp) -> (G GrpHom H) = S)
3	2	eleq2d 1544	. 2 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> F e. S))
4		elisset 1820	. . . . 5 |- (F e. S -> F e. V)
5		feq1 3626	. . . . . . . 8 |- (f = F -> (f:ran G-->ran H <-> F:ran G-->ran H))
6		fveq1 3729	. . . . . . . . . . 11 |- (f = F -> (f` x) = (F` x))
7		fveq1 3729	. . . . . . . . . . 11 |- (f = F -> (f` y) = (F` y))
8	6, 7	opreq12d 3984	. . . . . . . . . 10 |- (f = F -> ((f` x)H(f` y)) = ((F` x)H(F` y)))
9		fveq1 3729	. . . . . . . . . 10 |- (f = F -> (f` (xGy)) = (F` (xGy)))
10	8, 9	eqeq12d 1492	. . . . . . . . 9 |- (f = F -> (((f` x)H(f` y)) = (f` (xGy)) <-> ((F` x)H(F` y)) = (F` (xGy))))
11	10	2ralbidv 1683	. . . . . . . 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 630	. . . . . . 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 1903	. . . . . 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 3365	. . . . . . 7 |- (G e. Grp -> ran G e. V)
17		fex 3658	. . . . . . . 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 393	. . . . 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 520	. . 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 391	. 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 530	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 958   e. wcel 960  {cab 1466  A.wral 1648  Vcvv 1814  ran crn 3177  -->wf 3184  ` cfv 3188  (class class class)co 3969  Grpcgr 8030   GrpHom cghom 10373

This theorem is referenced by:  elghom 10379

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-ghom 10375

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