Theorem gicer 19308

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 19291	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
2		cnvimass 6102	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3		gimfn 19292	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4	3	fndmi 6673	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
5	2, 4	sseqtri 4032	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
6	1, 5	eqsstri 4030	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
7		relxp 5707	. . 3 ⊢ Rel (Grp × Grp)
8		relss 5794	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃𝑔
10		gicsym 19306	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
11		gictr 19307	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
12		gicref 19303	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
13		giclcl 19304	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
14	12, 13	impbii 209	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
15	9, 10, 11, 14	iseri 8771	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963   class class class wbr 5148   × cxp 5687  ^◡ccnv 5688  dom cdm 5689   “ cima 5692  Rel wrel 5694  1_oc1o 8498   Er wer 8741  Grpcgrp 18964   GrpIso cgim 19288   ≃_𝑔 cgic 19289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244  df-gim 19290  df-gic 19291

This theorem is referenced by: (None)