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Theorem gidsn 34064
Description: Obsolete as of 23-Jan-2020. Use mnd1id 17533 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 33994 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3 opex 5081 . . . . 5 𝐴, 𝐴⟩ ∈ V
43rnsnop 5776 . . . 4 ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
54eqcomi 2769 . . 3 {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6 eqid 2760 . . 3 (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
75, 6grpoidcl 27677 . 2 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8 elsni 4338 . 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 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327  ran crn 5267  cfv 6049  GrpOpcgr 27652  GIdcgi 27653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-grpo 27656  df-gid 27657
This theorem is referenced by:  zrdivrng  34065
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