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Theorem gidsn 35696
 Description: Obsolete as of 23-Jan-2020. Use mnd1id 18024 instead. The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablsn.1 𝐴 ∈ V
Assertion
Ref Expression
gidsn (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴

Proof of Theorem gidsn
StepHypRef Expression
1 ablsn.1 . . 3 𝐴 ∈ V
21grposnOLD 35626 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3 opex 5327 . . . . 5 𝐴, 𝐴⟩ ∈ V
43rnsnop 6057 . . . 4 ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
54eqcomi 2767 . . 3 {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6 eqid 2758 . . 3 (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
75, 6grpoidcl 28401 . 2 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8 elsni 4542 . 2 ((GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴} → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴)
92, 7, 8mp2b 10 1 (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4525  ⟨cop 4531  ran crn 5528  ‘cfv 6339  GrpOpcgr 28376  GIdcgi 28377 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-grpo 28380  df-gid 28381 This theorem is referenced by:  zrdivrng  35697
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