gicer - Metamath Proof Explorer

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Theorem gicer 18408

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 18392	. . . 4 ⊢ ≃_𝑔 = (^◡ GrpIso “ (V ∖ 1_o))
2		cnvimass 5916	. . . . 5 ⊢ (^◡ GrpIso “ (V ∖ 1_o)) ⊆ dom GrpIso
3		gimfn 18393	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4	3	fndmi 6426	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
5	2, 4	sseqtri 3951	. . . 4 ⊢ (^◡ GrpIso “ (V ∖ 1_o)) ⊆ (Grp × Grp)
6	1, 5	eqsstri 3949	. . 3 ⊢ ≃_𝑔 ⊆ (Grp × Grp)
7		relxp 5537	. . 3 ⊢ Rel (Grp × Grp)
8		relss 5620	. . 3 ⊢ ( ≃_𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃_𝑔 ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃_𝑔
10		gicsym 18406	. 2 ⊢ (𝑥 ≃_𝑔 𝑦 → 𝑦 ≃_𝑔 𝑥)
11		gictr 18407	. 2 ⊢ ((𝑥 ≃_𝑔 𝑦 ∧ 𝑦 ≃_𝑔 𝑧) → 𝑥 ≃_𝑔 𝑧)
12		gicref 18403	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃_𝑔 𝑥)
13		giclcl 18404	. . 3 ⊢ (𝑥 ≃_𝑔 𝑥 → 𝑥 ∈ Grp)
14	12, 13	impbii 212	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃_𝑔 𝑥)
15	9, 10, 11, 14	iseri 8299	1 ⊢ ≃_𝑔 Er Grp

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 ^◡ccnv 5518 dom cdm 5519 “ cima 5522 Rel wrel 5524 1_oc1o 8078 Er wer 8269 Grpcgrp 18095 GrpIso cgim 18389 ≃_𝑔 cgic 18390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-1o 8085 df-er 8272 df-map 8391 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-gim 18391 df-gic 18392

This theorem is referenced by: (None)

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