Theorem elgiso 10398

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

Assertion

Ref	Expression
elgiso	|- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpIso H) <-> (F e. (G GrpHom H) /\ F:ran G-1-1-onto->ran H)))

Proof of Theorem elgiso

Step	Hyp	Ref	Expression
1		oprex 3983	. . . . 5 |- (G GrpHom H) e. V
2	1	rabex 2725	. . . 4 |- {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H} e. V
3		rneq 3339	. . . . . . 7 |- (g = G -> ran g = ran G)
4		f1oeq2 3685	. . . . . . 7 |- (ran g = ran G -> (f:ran g-1-1-onto->ran h <-> f:ran G-1-1-onto->ran h))
5	3, 4	syl 10	. . . . . 6 |- (g = G -> (f:ran g-1-1-onto->ran h <-> f:ran G-1-1-onto->ran h))
6	5	rabbisdv 1807	. . . . 5 |- (g = G -> {f e. (g GrpHom h) | f:ran g-1-1-onto->ran h} = {f e. (g GrpHom h) | f:ran G-1-1-onto->ran h})
7		opreq1 3968	. . . . . 6 |- (g = G -> (g GrpHom h) = (G GrpHom h))
8		rabeq 1809	. . . . . 6 |- ((g GrpHom h) = (G GrpHom h) -> {f e. (g GrpHom h) | f:ran G-1-1-onto->ran h} = {f e. (G GrpHom h) | f:ran G-1-1-onto->ran h})
9	7, 8	syl 10	. . . . 5 |- (g = G -> {f e. (g GrpHom h) | f:ran G-1-1-onto->ran h} = {f e. (G GrpHom h) | f:ran G-1-1-onto->ran h})
10	6, 9	eqtrd 1507	. . . 4 |- (g = G -> {f e. (g GrpHom h) | f:ran g-1-1-onto->ran h} = {f e. (G GrpHom h) | f:ran G-1-1-onto->ran h})
11		rneq 3339	. . . . . . 7 |- (h = H -> ran h = ran H)
12		f1oeq3 3686	. . . . . . 7 |- (ran h = ran H -> (f:ran G-1-1-onto->ran h <-> f:ran G-1-1-onto->ran H))
13	11, 12	syl 10	. . . . . 6 |- (h = H -> (f:ran G-1-1-onto->ran h <-> f:ran G-1-1-onto->ran H))
14	13	rabbisdv 1807	. . . . 5 |- (h = H -> {f e. (G GrpHom h) | f:ran G-1-1-onto->ran h} = {f e. (G GrpHom h) | f:ran G-1-1-onto->ran H})
15		opreq2 3969	. . . . . 6 |- (h = H -> (G GrpHom h) = (G GrpHom H))
16		rabeq 1809	. . . . . 6 |- ((G GrpHom h) = (G GrpHom H) -> {f e. (G GrpHom h) | f:ran G-1-1-onto->ran H} = {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H})
17	15, 16	syl 10	. . . . 5 |- (h = H -> {f e. (G GrpHom h) | f:ran G-1-1-onto->ran H} = {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H})
18	14, 17	eqtrd 1507	. . . 4 |- (h = H -> {f e. (G GrpHom h) | f:ran G-1-1-onto->ran h} = {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H})
19		df-giso 10381	. . . 4 |- GrpIso = {<.<.g, h>., s>. | ((g e. Grp /\ h e. Grp) /\ s = {f e. (g GrpHom h) | f:ran g-1-1-onto->ran h})}
20	2, 10, 18, 19	oprabval2 4028	. . 3 |- ((G e. Grp /\ H e. Grp) -> (G GrpIso H) = {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H})
21	20	eleq2d 1541	. 2 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpIso H) <-> F e. {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H}))
22		f1oeq1 3684	. . 3 |- (f = F -> (f:ran G-1-1-onto->ran H <-> F:ran G-1-1-onto->ran H))
23	22	elrab 1905	. 2 |- (F e. {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H} <-> (F e. (G GrpHom H) /\ F:ran G-1-1-onto->ran H))
24	21, 23	syl6bb 536	1 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpIso H) <-> (F e. (G GrpHom H) /\ F:ran G-1-1-onto->ran H)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  ran crn 3171  -1-1-onto->wf1o 3181  (class class class)co 3963  Grpcgr 8033   GrpHom cghom 10378   GrpIso cgiso 10379

This theorem is referenced by:  cayleylem3 10411

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-rex 1650  df-rab 1652  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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-giso 10381

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