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Theorem gicer 18408
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 18392 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1o))
2 cnvimass 5942 . . . . 5 ( GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3 gimfn 18393 . . . . . 6 GrpIso Fn (Grp × Grp)
4 fndm 6448 . . . . . 6 ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
53, 4ax-mp 5 . . . . 5 dom GrpIso = (Grp × Grp)
62, 5sseqtri 4001 . . . 4 ( GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
71, 6eqsstri 3999 . . 3 𝑔 ⊆ (Grp × Grp)
8 relxp 5566 . . 3 Rel (Grp × Grp)
9 relss 5649 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
107, 8, 9mp2 9 . 2 Rel ≃𝑔
11 gicsym 18406 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
12 gictr 18407 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
13 gicref 18403 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
14 giclcl 18404 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1513, 14impbii 211 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
1610, 11, 12, 15iseri 8308 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1531  wcel 2108  Vcvv 3493  cdif 3931  wss 3934   class class class wbr 5057   × cxp 5546  ccnv 5547  dom cdm 5548  cima 5551  Rel wrel 5553   Fn wfn 6343  1oc1o 8087   Er wer 8278  Grpcgrp 18095   GrpIso cgim 18389  𝑔 cgic 18390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-1o 8094  df-er 8281  df-map 8400  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-gim 18391  df-gic 18392
This theorem is referenced by: (None)
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