Theorem gicer 19265

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 19248	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
2		cnvimass 6074	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3		gimfn 19249	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4	3	fndmi 6647	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
5	2, 4	sseqtri 4012	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
6	1, 5	eqsstri 4010	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
7		relxp 5677	. . 3 ⊢ Rel (Grp × Grp)
8		relss 5765	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃𝑔
10		gicsym 19263	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
11		gictr 19264	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
12		gicref 19260	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
13		giclcl 19261	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
14	12, 13	impbii 209	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
15	9, 10, 11, 14	iseri 8751	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931   class class class wbr 5124   × cxp 5657  ^◡ccnv 5658  dom cdm 5659   “ cima 5662  Rel wrel 5664  1_oc1o 8478   Er wer 8721  Grpcgrp 18921   GrpIso cgim 19245   ≃_𝑔 cgic 19246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-1o 8485  df-er 8724  df-map 8847  df-0g 17460  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-grp 18924  df-ghm 19201  df-gim 19247  df-gic 19248

This theorem is referenced by: (None)