Theorem gicer 19348

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 19331	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
2		cnvimass 6086	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3		gimfn 19332	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4	3	fndmi 6641	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
5	2, 4	sseqtri 3986	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
6	1, 5	eqsstri 3984	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
7		relxp 5681	. . 3 ⊢ Rel (Grp × Grp)
8		relss 5770	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃𝑔
10		gicsym 19346	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
11		gictr 19347	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
12		gicref 19343	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
13		giclcl 19344	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
14	12, 13	impbii 212	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
15	9, 10, 11, 14	iseri 8723	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906   class class class wbr 5110   × cxp 5661  ^◡ccnv 5662  dom cdm 5663   “ cima 5666  Rel wrel 5668  1_oc1o 8447   Er wer 8692  Grpcgrp 19001   GrpIso cgim 19328   ≃_𝑔 cgic 19329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-1o 8454  df-er 8695  df-map 8827  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-grp 19004  df-ghm 19285  df-gim 19330  df-gic 19331

This theorem is referenced by: (None)