Theorem elgiso 10393

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 3989	. . . . 5 |- (G GrpHom H) e. V
2	1	rabex 2730	. . . 4 |- {f e. (G GrpHom H) | f:ran G-1-1-onto->ran H} e. V
3		rneq 3345	. . . . . . 7 |- (g = G -> ran g = ran G)
4		f1oeq2 3691	. . . . . . 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 1810	. . . . 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 3974	. . . . . 6 |- (g = G -> (g GrpHom h) = (G GrpHom h))
8		rabeq 1812	. . . . . 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 1510	. . . 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 3345	. . . . . . 7 |- (h = H -> ran h = ran H)
12		f1oeq3 3692	. . . . . . 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 1810	. . . . 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 3975	. . . . . 6 |- (h = H -> (G GrpHom h) = (G GrpHom H))
16		rabeq 1812	. . . . . 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 1510	. . . 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 10376	. . . 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 4034	. . 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 1544	. 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 3690	. . 3 |- (f = F -> (f:ran G-1-1-onto->ran H <-> F:ran G-1-1-onto->ran H))
23	22	elrab 1908	. 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 538	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 958   e. wcel 960  {crab 1651  ran crn 3177  -1-1-onto->wf1o 3187  (class class class)co 3969  Grpcgr 8030   GrpHom cghom 10373   GrpIso cgiso 10374

This theorem is referenced by:  cayleylem3 10406

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-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-rex 1653  df-rab 1655  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-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-oprab 3972  df-giso 10376

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