Theorem isgim 15901

Description: An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses

Ref	Expression
isgim.b	|- B = ( Base ` R )
isgim.c	|- C = ( Base ` S )

Assertion

Ref	Expression
isgim	|- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) )

Proof of Theorem isgim

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 967	. 2 |- ( ( R e. Grp /\ S e. Grp /\ F e. { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } ) <-> ( ( R e. Grp /\ S e. Grp ) /\ F e. { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } ) )
2		df-gim 15898	. . 3 |- GrpIso = ( a e. Grp , b e. Grp |-> { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } )
3		ovex 6218	. . . 4 |- ( a GrpHom b ) e. _V
4	3	rabex 4544	. . 3 |- { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } e. _V
5		oveq12 6202	. . . 4 |- ( ( a = R /\ b = S ) -> ( a GrpHom b ) = ( R GrpHom S ) )
6		fveq2 5792	. . . . . 6 |- ( a = R -> ( Base ` a ) = ( Base ` R ) )
7		isgim.b	. . . . . 6 |- B = ( Base ` R )
8	6, 7	syl6eqr 2510	. . . . 5 |- ( a = R -> ( Base ` a ) = B )
9		fveq2 5792	. . . . . 6 |- ( b = S -> ( Base ` b ) = ( Base ` S ) )
10		isgim.c	. . . . . 6 |- C = ( Base ` S )
11	9, 10	syl6eqr 2510	. . . . 5 |- ( b = S -> ( Base ` b ) = C )
12		f1oeq23 5736	. . . . 5 |- ( ( ( Base ` a ) = B /\ ( Base ` b ) = C ) -> ( c : ( Base ` a ) -1-1-onto-> ( Base ` b ) <-> c : B -1-1-onto-> C ) )
13	8, 11, 12	syl2an 477	. . . 4 |- ( ( a = R /\ b = S ) -> ( c : ( Base ` a ) -1-1-onto-> ( Base ` b ) <-> c : B -1-1-onto-> C ) )
14	5, 13	rabeqbidv 3066	. . 3 |- ( ( a = R /\ b = S ) -> { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } = { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } )
15	2, 4, 14	elovmpt2 6410	. 2 |- ( F e. ( R GrpIso S ) <-> ( R e. Grp /\ S e. Grp /\ F e. { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } ) )
16		ghmgrp1 15860	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
17		ghmgrp2 15861	. . . . . 6 |- ( F e. ( R GrpHom S ) -> S e. Grp )
18	16, 17	jca 532	. . . . 5 |- ( F e. ( R GrpHom S ) -> ( R e. Grp /\ S e. Grp ) )
19	18	adantr 465	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) -> ( R e. Grp /\ S e. Grp ) )
20	19	pm4.71ri 633	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) <-> ( ( R e. Grp /\ S e. Grp ) /\ ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) ) )
21		f1oeq1 5733	. . . . 5 |- ( c = F -> ( c : B -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
22	21	elrab 3217	. . . 4 |- ( F e. { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } <-> ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) )
23	22	anbi2i 694	. . 3 |- ( ( ( R e. Grp /\ S e. Grp ) /\ F e. { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } ) <-> ( ( R e. Grp /\ S e. Grp ) /\ ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) ) )
24	20, 23	bitr4i 252	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) <-> ( ( R e. Grp /\ S e. Grp ) /\ F e. { c e. ( R GrpHom S ) | c : B -1-1-onto-> C } ) )
25	1, 15, 24	3bitr4i 277	1 |- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) )

Syntax hints:   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   {crab 2799   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   Basecbs 14285   Grpcgrp 15521   GrpHom cghm 15855   GrpIso cgim 15896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-ghm 15856  df-gim 15898

This theorem is referenced by:  gimf1o  15902  gimghm  15903  isgim2  15904  invoppggim  15986  lmimgim  17261  zzngim  18103  cygznlem3  18120  reefgim  22041  imasgim  29596  pmattomply1grpiso  31274