Theorem isgim 16184

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 976	. 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 16181	. . 3 |- GrpIso = ( a e. Grp , b e. Grp |-> { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } )
3		ovex 6309	. . . 4 |- ( a GrpHom b ) e. _V
4	3	rabex 4588	. . 3 |- { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } e. _V
5		oveq12 6290	. . . 4 |- ( ( a = R /\ b = S ) -> ( a GrpHom b ) = ( R GrpHom S ) )
6		fveq2 5856	. . . . . 6 |- ( a = R -> ( Base ` a ) = ( Base ` R ) )
7		isgim.b	. . . . . 6 |- B = ( Base ` R )
8	6, 7	syl6eqr 2502	. . . . 5 |- ( a = R -> ( Base ` a ) = B )
9		fveq2 5856	. . . . . 6 |- ( b = S -> ( Base ` b ) = ( Base ` S ) )
10		isgim.c	. . . . . 6 |- C = ( Base ` S )
11	9, 10	syl6eqr 2502	. . . . 5 |- ( b = S -> ( Base ` b ) = C )
12		f1oeq23 5800	. . . . 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 3090	. . 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 6505	. 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 16143	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
17		ghmgrp2 16144	. . . . . 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 5797	. . . . 5 |- ( c = F -> ( c : B -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
22	21	elrab 3243	. . . 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 974   = wceq 1383   e. wcel 1804   {crab 2797   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   Basecbs 14509   Grpcgrp 15927   GrpHom cghm 16138   GrpIso cgim 16179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-ghm 16139  df-gim 16181

This theorem is referenced by:  gimf1o  16185  gimghm  16186  isgim2  16187  invoppggim  16269  rimgim  17259  lmimgim  17585  zzngim  18464  cygznlem3  18481  pm2mpgrpiso  19191  reefgim  22717  imasgim  31023