Theorem elghom 10375

Description: Membership in the set of group homomorphisms from G to H. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
elghom.1	|- X = ran G
elghom.2	|- W = ran H

Assertion

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

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

Proof of Theorem elghom

Step	Hyp	Ref	Expression
1		eqid 1475	. . 3 |- {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)))}
2	1	elghomlem2 10374	. 2 |- ((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)))))
3		elghom.1	. . . 4 |- X = ran G
4		elghom.2	. . . 4 |- W = ran H
5		feq23 3620	. . . 4 |- ((X = ran G /\ W = ran H) -> (F:X-->W <-> F:ran G-->ran H))
6	3, 4, 5	mp2an 696	. . 3 |- (F:X-->W <-> F:ran G-->ran H)
7		raleq1 1785	. . . . . 6 |- (X = ran G -> (A.y e. X ((F` x)H(F` y)) = (F` (xGy)) <-> A.y e. ran G((F` x)H(F` y)) = (F` (xGy))))
8	3, 7	ax-mp 7	. . . . 5 |- (A.y e. X ((F` x)H(F` y)) = (F` (xGy)) <-> A.y e. ran G((F` x)H(F` y)) = (F` (xGy)))
9	8	ralbii 1666	. . . 4 |- (A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)) <-> A.x e. X A.y e. ran G((F` x)H(F` y)) = (F` (xGy)))
10		raleq1 1785	. . . . 5 |- (X = ran G -> (A.x e. X A.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))))
11	3, 10	ax-mp 7	. . . 4 |- (A.x e. X A.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	9, 11	bitr 173	. . 3 |- (A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)) <-> A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))
13	6, 12	anbi12i 482	. 2 |- ((F:X-->W /\ A.x e. X A.y e. X ((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))))
14	2, 13	syl6bbr 537	1 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:X-->W /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  {cab 1463  A.wral 1644  ran crn 3168  -->wf 3175  ` cfv 3179  (class class class)co 3960  Grpcgr 8016   GrpHom cghom 10369

This theorem is referenced by:  ghomgrpilem1 10376  ghomgrpilem2 10377  ghomsn 10379  ghomfo 10382  ghomlin 10384  ghomid 10385  ghomgsg 10386  cayleylem2 10401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-oprab 3963  df-ghom 10371

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