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Theorem gicer 18527
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 18511 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1o))
2 cnvimass 5917 . . . . 5 ( GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3 gimfn 18512 . . . . . 6 GrpIso Fn (Grp × Grp)
43fndmi 6435 . . . . 5 dom GrpIso = (Grp × Grp)
52, 4sseqtri 3911 . . . 4 ( GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
61, 5eqsstri 3909 . . 3 𝑔 ⊆ (Grp × Grp)
7 relxp 5537 . . 3 Rel (Grp × Grp)
8 relss 5621 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
96, 7, 8mp2 9 . 2 Rel ≃𝑔
10 gicsym 18525 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
11 gictr 18526 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
12 gicref 18522 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
13 giclcl 18523 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1412, 13impbii 212 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
159, 10, 11, 14iseri 8340 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:  wcel 2113  Vcvv 3397  cdif 3838  wss 3841   class class class wbr 5027   × cxp 5517  ccnv 5518  dom cdm 5519  cima 5522  Rel wrel 5524  1oc1o 8117   Er wer 8310  Grpcgrp 18212   GrpIso cgim 18508  𝑔 cgic 18509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-1o 8124  df-er 8313  df-map 8432  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-mhm 18065  df-grp 18215  df-ghm 18467  df-gim 18510  df-gic 18511
This theorem is referenced by: (None)
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