Theorem gicer 17541

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 17525	. . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1𝑜))
2		cnvimass 5404	. . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ dom GrpIso
3		gimfn 17526	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4		fndm 5904	. . . . . 6 ⊢ ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
5	3, 4	ax-mp 5	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
6	2, 5	sseqtri 3600	. . . 4 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ (Grp × Grp)
7	1, 6	eqsstri 3598	. . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp)
8		relxp 5150	. . 3 ⊢ Rel (Grp × Grp)
9		relss 5129	. . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
10	7, 8, 9	mp2 9	. 2 ⊢ Rel ≃𝑔
11		gicsym 17539	. 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
12		gictr 17540	. 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
13		gicref 17536	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
14		giclcl 17537	. . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
15	13, 14	impbii 198	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
16	10, 11, 12, 15	iseri 7656	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Rel wrel 5043   Fn wfn 5799  1_𝑜c1o 7440   Er wer 7626  Grpcgrp 17245   GrpIso cgim 17522   ≃_𝑔 cgic 17523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-gim 17524  df-gic 17525

This theorem is referenced by: (None)