Theorem gidsn 37959

Description: Obsolete as of 23-Jan-2020. Use mnd1id 18793 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

Step	Hyp	Ref	Expression
1		ablsn.1	. . 3 ⊢ 𝐴 ∈ V
2	1	grposnOLD 37889	. 2 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3		opex 5469	. . . . 5 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
4	3	rnsnop 6244	. . . 4 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
5	4	eqcomi 2746	. . 3 ⊢ {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6		eqid 2737	. . 3 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
7	5, 6	grpoidcl 30533	. 2 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8		elsni 4643	. 2 ⊢ ((GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴} → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴)
9	2, 7, 8	mp2b 10	1 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  ⟨cop 4632  ran crn 5686  ‘cfv 6561  GrpOpcgr 30508  GIdcgi 30509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-grpo 30512  df-gid 30513

This theorem is referenced by:  zrdivrng  37960