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Theorem gicer 19295
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 19278 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1o))
2 cnvimass 6100 . . . . 5 ( GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3 gimfn 19279 . . . . . 6 GrpIso Fn (Grp × Grp)
43fndmi 6672 . . . . 5 dom GrpIso = (Grp × Grp)
52, 4sseqtri 4032 . . . 4 ( GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
61, 5eqsstri 4030 . . 3 𝑔 ⊆ (Grp × Grp)
7 relxp 5703 . . 3 Rel (Grp × Grp)
8 relss 5791 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
96, 7, 8mp2 9 . 2 Rel ≃𝑔
10 gicsym 19293 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
11 gictr 19294 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
12 gicref 19290 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
13 giclcl 19291 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1412, 13impbii 209 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
159, 10, 11, 14iseri 8772 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  cdif 3948  wss 3951   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688  Rel wrel 5690  1oc1o 8499   Er wer 8742  Grpcgrp 18951   GrpIso cgim 19275  𝑔 cgic 19276
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-ghm 19231  df-gim 19277  df-gic 19278
This theorem is referenced by: (None)
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