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Theorem gicer 19067
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 19051 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1o))
2 cnvimass 6034 . . . . 5 ( GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3 gimfn 19052 . . . . . 6 GrpIso Fn (Grp × Grp)
43fndmi 6607 . . . . 5 dom GrpIso = (Grp × Grp)
52, 4sseqtri 3981 . . . 4 ( GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
61, 5eqsstri 3979 . . 3 𝑔 ⊆ (Grp × Grp)
7 relxp 5652 . . 3 Rel (Grp × Grp)
8 relss 5738 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
96, 7, 8mp2 9 . 2 Rel ≃𝑔
10 gicsym 19065 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
11 gictr 19066 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
12 gicref 19062 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
13 giclcl 19063 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1412, 13impbii 208 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
159, 10, 11, 14iseri 8676 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3446  cdif 3908  wss 3911   class class class wbr 5106   × cxp 5632  ccnv 5633  dom cdm 5634  cima 5637  Rel wrel 5639  1oc1o 8406   Er wer 8646  Grpcgrp 18749   GrpIso cgim 19048  𝑔 cgic 19049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-1o 8413  df-er 8649  df-map 8768  df-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-mhm 18602  df-grp 18752  df-ghm 19007  df-gim 19050  df-gic 19051
This theorem is referenced by: (None)
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