Theorem gicer 18031

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 18015	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1𝑜))
2		cnvimass 5702	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ dom GrpIso
3		gimfn 18016	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4		fndm 6201	. . . . . 6 ⊢ ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
5	3, 4	ax-mp 5	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
6	2, 5	sseqtri 3833	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ (Grp × Grp)
7	1, 6	eqsstri 3831	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
8		relxp 5330	. . 3 ⊢ Rel (Grp × Grp)
9		relss 5411	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
10	7, 8, 9	mp2 9	. 2 ⊢ Rel ≃𝑔
11		gicsym 18029	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
12		gictr 18030	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
13		gicref 18026	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
14		giclcl 18027	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
15	13, 14	impbii 201	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
16	10, 11, 12, 15	iseri 8009	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   = wceq 1653   ∈ wcel 2157  Vcvv 3385   ∖ cdif 3766   ⊆ wss 3769   class class class wbr 4843   × cxp 5310  ^◡ccnv 5311  dom cdm 5312   “ cima 5315  Rel wrel 5317   Fn wfn 6096  1_𝑜c1o 7792   Er wer 7979  Grpcgrp 17738   GrpIso cgim 18012   ≃_𝑔 cgic 18013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-1o 7799  df-er 7982  df-map 8097  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-mhm 17650  df-grp 17741  df-ghm 17971  df-gim 18014  df-gic 18015

This theorem is referenced by: (None)