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Theorem gicer 17639
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 17623 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1𝑜))
2 cnvimass 5444 . . . . 5 ( GrpIso “ (V ∖ 1𝑜)) ⊆ dom GrpIso
3 gimfn 17624 . . . . . 6 GrpIso Fn (Grp × Grp)
4 fndm 5948 . . . . . 6 ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
53, 4ax-mp 5 . . . . 5 dom GrpIso = (Grp × Grp)
62, 5sseqtri 3616 . . . 4 ( GrpIso “ (V ∖ 1𝑜)) ⊆ (Grp × Grp)
71, 6eqsstri 3614 . . 3 𝑔 ⊆ (Grp × Grp)
8 relxp 5188 . . 3 Rel (Grp × Grp)
9 relss 5167 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
107, 8, 9mp2 9 . 2 Rel ≃𝑔
11 gicsym 17637 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
12 gictr 17638 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
13 gicref 17634 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
14 giclcl 17635 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1513, 14impbii 199 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
1610, 11, 12, 15iseri 7714 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  wss 3555   class class class wbr 4613   × cxp 5072  ccnv 5073  dom cdm 5074  cima 5077  Rel wrel 5079   Fn wfn 5842  1𝑜c1o 7498   Er wer 7684  Grpcgrp 17343   GrpIso cgim 17620  𝑔 cgic 17621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-1o 7505  df-er 7687  df-map 7804  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-ghm 17579  df-gim 17622  df-gic 17623
This theorem is referenced by: (None)
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