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Theorem gidsn 37664
Description: Obsolete as of 23-Jan-2020. Use mnd1id 18763 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 37594 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3 opex 5461 . . . . 5 𝐴, 𝐴⟩ ∈ V
43rnsnop 6226 . . . 4 ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
54eqcomi 2735 . . 3 {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6 eqid 2726 . . 3 (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
75, 6grpoidcl 30442 . 2 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8 elsni 4641 . 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 1534  wcel 2099  Vcvv 3463  {csn 4624  cop 4630  ran crn 5674  cfv 6544  GrpOpcgr 30417  GIdcgi 30418
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-grpo 30421  df-gid 30422
This theorem is referenced by:  zrdivrng  37665
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