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Theorem gidsn 35117
Description: Obsolete as of 23-Jan-2020. Use mnd1id 17948 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 35047 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3 opex 5353 . . . . 5 𝐴, 𝐴⟩ ∈ V
43rnsnop 6080 . . . 4 ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
54eqcomi 2835 . . 3 {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6 eqid 2826 . . 3 (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
75, 6grpoidcl 28224 . 2 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8 elsni 4581 . 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 1530  wcel 2107  Vcvv 3500  {csn 4564  cop 4570  ran crn 5555  cfv 6354  GrpOpcgr 28199  GIdcgi 28200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-grpo 28203  df-gid 28204
This theorem is referenced by:  zrdivrng  35118
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