Theorem isgim 17704

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 1039	. 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 17701	. . 3 |- GrpIso = ( a e. Grp , b e. Grp |-> { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } )
3		ovex 6678	. . . 4 |- ( a GrpHom b ) e. _V
4	3	rabex 4813	. . 3 |- { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } e. _V
5		oveq12 6659	. . . 4 |- ( ( a = R /\ b = S ) -> ( a GrpHom b ) = ( R GrpHom S ) )
6		fveq2 6191	. . . . . 6 |- ( a = R -> ( Base ` a ) = ( Base ` R ) )
7		isgim.b	. . . . . 6 |- B = ( Base ` R )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( a = R -> ( Base ` a ) = B )
9		fveq2 6191	. . . . . 6 |- ( b = S -> ( Base ` b ) = ( Base ` S ) )
10		isgim.c	. . . . . 6 |- C = ( Base ` S )
11	9, 10	syl6eqr 2674	. . . . 5 |- ( b = S -> ( Base ` b ) = C )
12		f1oeq23 6130	. . . . 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 494	. . . 4 |- ( ( a = R /\ b = S ) -> ( c : ( Base ` a ) -1-1-onto-> ( Base ` b ) <-> c : B -1-1-onto-> C ) )
14	5, 13	rabeqbidv 3195	. . 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 6879	. 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 17662	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
17		ghmgrp2 17663	. . . . . 6 |- ( F e. ( R GrpHom S ) -> S e. Grp )
18	16, 17	jca 554	. . . . 5 |- ( F e. ( R GrpHom S ) -> ( R e. Grp /\ S e. Grp ) )
19	18	adantr 481	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) -> ( R e. Grp /\ S e. Grp ) )
20	19	pm4.71ri 665	. . 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 6127	. . . . 5 |- ( c = F -> ( c : B -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
22	21	elrab 3363	. . . 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 730	. . 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 267	. 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 292	1 |- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Grpcgrp 17422   GrpHom cghm 17657   GrpIso cgim 17699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ghm 17658  df-gim 17701

This theorem is referenced by:  gimf1o  17705  gimghm  17706  isgim2  17707  invoppggim  17790  rimgim  18736  lmimgim  19065  zzngim  19901  cygznlem3  19918  pm2mpgrpiso  20622  reefgim  24204  imasgim  37670