Theorem isgim 16098

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 970	. 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 16095	. . 3 |- GrpIso = ( a e. Grp , b e. Grp |-> { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } )
3		ovex 6300	. . . 4 |- ( a GrpHom b ) e. _V
4	3	rabex 4591	. . 3 |- { c e. ( a GrpHom b ) | c : ( Base ` a ) -1-1-onto-> ( Base ` b ) } e. _V
5		oveq12 6284	. . . 4 |- ( ( a = R /\ b = S ) -> ( a GrpHom b ) = ( R GrpHom S ) )
6		fveq2 5857	. . . . . 6 |- ( a = R -> ( Base ` a ) = ( Base ` R ) )
7		isgim.b	. . . . . 6 |- B = ( Base ` R )
8	6, 7	syl6eqr 2519	. . . . 5 |- ( a = R -> ( Base ` a ) = B )
9		fveq2 5857	. . . . . 6 |- ( b = S -> ( Base ` b ) = ( Base ` S ) )
10		isgim.c	. . . . . 6 |- C = ( Base ` S )
11	9, 10	syl6eqr 2519	. . . . 5 |- ( b = S -> ( Base ` b ) = C )
12		f1oeq23 5801	. . . . 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 3101	. . 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 6495	. 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 16057	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
17		ghmgrp2 16058	. . . . . 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 5798	. . . . 5 |- ( c = F -> ( c : B -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
22	21	elrab 3254	. . . 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 968   = wceq 1374   e. wcel 1762   {crab 2811   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   Basecbs 14479   Grpcgrp 15716   GrpHom cghm 16052   GrpIso cgim 16093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-ghm 16053  df-gim 16095

This theorem is referenced by:  gimf1o  16099  gimghm  16100  isgim2  16101  invoppggim  16183  rimgim  17161  lmimgim  17487  zzngim  18351  cygznlem3  18368  pm2mpgrpiso  19078  reefgim  22572  imasgim  30505