Theorem gicer 17939

Description: Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)

Assertion

Ref	Expression
gicer	⊢ ≃𝑔 Er Grp

Proof of Theorem gicer

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

Step	Hyp	Ref	Expression
1		df-gic 17923	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1𝑜))
2		cnvimass 5643	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ dom GrpIso
3		gimfn 17924	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4		fndm 6151	. . . . . 6 ⊢ ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
5	3, 4	ax-mp 5	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
6	2, 5	sseqtri 3778	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ (Grp × Grp)
7	1, 6	eqsstri 3776	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
8		relxp 5283	. . 3 ⊢ Rel (Grp × Grp)
9		relss 5363	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
10	7, 8, 9	mp2 9	. 2 ⊢ Rel ≃𝑔
11		gicsym 17937	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
12		gictr 17938	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
13		gicref 17934	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
14		giclcl 17935	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
15	13, 14	impbii 199	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
16	10, 11, 12, 15	iseri 7940	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712   ⊆ wss 3715   class class class wbr 4804   × cxp 5264  ^◡ccnv 5265  dom cdm 5266   “ cima 5269  Rel wrel 5271   Fn wfn 6044  1_𝑜c1o 7723   Er wer 7910  Grpcgrp 17643   GrpIso cgim 17920   ≃_𝑔 cgic 17921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-1o 7730  df-er 7913  df-map 8027  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-grp 17646  df-ghm 17879  df-gim 17922  df-gic 17923

This theorem is referenced by: (None)