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Theorem gidsn 34079
Description: Obsolete as of 23-Jan-2020. Use mnd1id 17539 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 34009 . 2 {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3 opex 5060 . . . . 5 𝐴, 𝐴⟩ ∈ V
43rnsnop 5759 . . . 4 ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
54eqcomi 2780 . . 3 {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6 eqid 2771 . . 3 (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
75, 6grpoidcl 27705 . 2 ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8 elsni 4333 . 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 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322  ran crn 5250  cfv 6031  GrpOpcgr 27680  GIdcgi 27681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-grpo 27684  df-gid 27685
This theorem is referenced by:  zrdivrng  34080
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