Theorem isgim 15781

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 15778	. . 3 |- GrpIso = ( a e. Grp , b e. Grp |-> { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } )
3		ovex 6111	. . . 4 |- ( a GrpHom b ) e. _V
4	3	rabex 4438	. . 3 |- { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } e. _V
5		oveq12 6095	. . . 4 |- ( ( a = R /\ b = S ) -> ( a GrpHom b ) = ( R GrpHom S ) )
6		fveq2 5686	. . . . . 6 |- ( a = R -> ( Base ` a ) = ( Base ` R ) )
7		isgim.b	. . . . . 6 |- B = ( Base ` R )
8	6, 7	syl6eqr 2488	. . . . 5 |- ( a = R -> ( Base ` a ) = B )
9		fveq2 5686	. . . . . 6 |- ( b = S -> ( Base ` b ) = ( Base ` S ) )
10		isgim.c	. . . . . 6 |- C = ( Base ` S )
11	9, 10	syl6eqr 2488	. . . . 5 |- ( b = S -> ( Base ` b ) = C )
12		f1oeq23 5630	. . . . 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 2962	. . 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 6302	. 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 15740	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
17		ghmgrp2 15741	. . . . . 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 5627	. . . . 5 |- ( c = F -> ( c : B -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
22	21	elrab 3112	. . . 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 1369   e. wcel 1756   {crab 2714   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Basecbs 14166   Grpcgrp 15402   GrpHom cghm 15735   GrpIso cgim 15776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-ghm 15736  df-gim 15778

This theorem is referenced by:  gimf1o  15782  gimghm  15783  isgim2  15784  invoppggim  15866  lmimgim  17123  zzngim  17960  cygznlem3  17977  reefgim  21890  imasgim  29408