Theorem gicer 19471

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 19454	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
2		cnvimass 6076	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3		gimfn 19455	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4	3	fndmi 6635	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
5	2, 4	sseqtri 3979	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
6	1, 5	eqsstri 3977	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
7		relxp 5669	. . 3 ⊢ Rel (Grp × Grp)
8		relss 5758	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃𝑔
10		gicsym 19469	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
11		gictr 19470	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
12		gicref 19466	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
13		giclcl 19467	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
14	12, 13	impbii 212	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
15	9, 10, 11, 14	iseri 8729	1 ⊢ ≃𝑔 Er Grp

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899   class class class wbr 5103   × cxp 5649  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  Rel wrel 5656  1_oc1o 8453   Er wer 8698  Grpcgrp 19124   GrpIso cgim 19451   ≃_𝑔 cgic 19452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-1o 8460  df-er 8701  df-map 8833  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-grp 19127  df-ghm 19408  df-gim 19453  df-gic 19454

This theorem is used by: (None)