Theorem gimcnv 17709

Description: The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
gimcnv	|- ( F e. ( S GrpIso T ) -> `' F e. ( T GrpIso S ) )

Proof of Theorem gimcnv

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
2		eqid 2622	. . . . . . 7 |- ( Base ` T ) = ( Base ` T )
3	1, 2	ghmf 17664	. . . . . 6 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
4		frel 6050	. . . . . . 7 |- ( F : ( Base ` S ) --> ( Base ` T ) -> Rel F )
5		dfrel2 5583	. . . . . . 7 |- ( Rel F <-> `' `' F = F )
6	4, 5	sylib 208	. . . . . 6 |- ( F : ( Base ` S ) --> ( Base ` T ) -> `' `' F = F )
7	3, 6	syl 17	. . . . 5 |- ( F e. ( S GrpHom T ) -> `' `' F = F )
8		id 22	. . . . 5 |- ( F e. ( S GrpHom T ) -> F e. ( S GrpHom T ) )
9	7, 8	eqeltrd 2701	. . . 4 |- ( F e. ( S GrpHom T ) -> `' `' F e. ( S GrpHom T ) )
10	9	anim2i 593	. . 3 |- ( ( `' F e. ( T GrpHom S ) /\ F e. ( S GrpHom T ) ) -> ( `' F e. ( T GrpHom S ) /\ `' `' F e. ( S GrpHom T ) ) )
11	10	ancoms 469	. 2 |- ( ( F e. ( S GrpHom T ) /\ `' F e. ( T GrpHom S ) ) -> ( `' F e. ( T GrpHom S ) /\ `' `' F e. ( S GrpHom T ) ) )
12		isgim2 17707	. 2 |- ( F e. ( S GrpIso T ) <-> ( F e. ( S GrpHom T ) /\ `' F e. ( T GrpHom S ) ) )
13		isgim2 17707	. 2 |- ( `' F e. ( T GrpIso S ) <-> ( `' F e. ( T GrpHom S ) /\ `' `' F e. ( S GrpHom T ) ) )
14	11, 12, 13	3imtr4i 281	1 |- ( F e. ( S GrpIso T ) -> `' F e. ( T GrpIso S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   `'ccnv 5113   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-gim 17701

This theorem is referenced by:  gicsym  17716  reloggim  24345  abliso  29696