Theorem gicerOLD 17719

Description: Obsolete proof of gicer 17718 as of 1-May-2021. Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
gicerOLD	|- ~=g𝑔 Er Grp

Proof of Theorem gicerOLD

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-gic 17702	. . . . . 6 |- ~=g𝑔 = ( `' GrpIso " ( _V \ 1o ) )
2		cnvimass 5485	. . . . . . 7 |- ( `' GrpIso " ( _V \ 1o ) ) C_ dom GrpIso
3		gimfn 17703	. . . . . . . 8 |- GrpIso Fn ( Grp X. Grp )
4		fndm 5990	. . . . . . . 8 |- ( GrpIso Fn ( Grp X. Grp ) -> dom GrpIso = ( Grp X. Grp ) )
5	3, 4	ax-mp 5	. . . . . . 7 |- dom GrpIso = ( Grp X. Grp )
6	2, 5	sseqtri 3637	. . . . . 6 |- ( `' GrpIso " ( _V \ 1o ) ) C_ ( Grp X. Grp )
7	1, 6	eqsstri 3635	. . . . 5 |- ~=g𝑔 C_ ( Grp X. Grp )
8		relxp 5227	. . . . 5 |- Rel ( Grp X. Grp )
9		relss 5206	. . . . 5 |- ( ~=g𝑔 C_ ( Grp X. Grp ) -> ( Rel ( Grp X. Grp ) -> Rel ~=g𝑔 ) )
10	7, 8, 9	mp2 9	. . . 4 |- Rel ~=g𝑔
11	10	a1i 11	. . 3 |- ( T. -> Rel ~=g𝑔 )
12		gicsym 17716	. . . 4 |- ( x ~=g𝑔 y -> y ~=g𝑔 x )
13	12	adantl 482	. . 3 |- ( ( T. /\ x ~=g𝑔 y ) -> y ~=g𝑔 x )
14		gictr 17717	. . . 4 |- ( ( x ~=g𝑔 y /\ y ~=g𝑔 z ) -> x ~=g𝑔 z )
15	14	adantl 482	. . 3 |- ( ( T. /\ ( x ~=g𝑔 y /\ y ~=g𝑔 z ) ) -> x ~=g𝑔 z )
16		gicref 17713	. . . . 5 |- ( x e. Grp -> x ~=g𝑔 x )
17		giclcl 17714	. . . . 5 |- ( x ~=g𝑔 x -> x e. Grp )
18	16, 17	impbii 199	. . . 4 |- ( x e. Grp <-> x ~=g𝑔 x )
19	18	a1i 11	. . 3 |- ( T. -> ( x e. Grp <-> x ~=g𝑔 x ) )
20	11, 13, 15, 19	iserd 7768	. 2 |- ( T. -> ~=g𝑔 Er Grp )
21	20	trud 1493	1 |- ~=g𝑔 Er Grp

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Rel wrel 5119   Fn wfn 5883   1oc1o 7553   Er wer 7739   Grpcgrp 17422   GrpIso cgim 17699   ~=g_𝑔 cgic 17700

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-rmo 2920  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-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-gim 17701  df-gic 17702

This theorem is referenced by: (None)