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Theorem gicer 19193
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.
StepHypRef Expression
1 df-gic 19176 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1o))
2 cnvimass 6081 . . . . 5 ( GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3 gimfn 19177 . . . . . 6 GrpIso Fn (Grp × Grp)
43fndmi 6654 . . . . 5 dom GrpIso = (Grp × Grp)
52, 4sseqtri 4019 . . . 4 ( GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
61, 5eqsstri 4017 . . 3 𝑔 ⊆ (Grp × Grp)
7 relxp 5695 . . 3 Rel (Grp × Grp)
8 relss 5782 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
96, 7, 8mp2 9 . 2 Rel ≃𝑔
10 gicsym 19191 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
11 gictr 19192 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
12 gicref 19188 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
13 giclcl 19189 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1412, 13impbii 208 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
159, 10, 11, 14iseri 8734 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:  wcel 2104  Vcvv 3472  cdif 3946  wss 3949   class class class wbr 5149   × cxp 5675  ccnv 5676  dom cdm 5677  cima 5680  Rel wrel 5682  1oc1o 8463   Er wer 8704  Grpcgrp 18857   GrpIso cgim 19173  𝑔 cgic 19174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7979  df-2nd 7980  df-1o 8470  df-er 8707  df-map 8826  df-0g 17393  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18707  df-grp 18860  df-ghm 19130  df-gim 19175  df-gic 19176
This theorem is referenced by: (None)
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