Theorem gidsn 38411

Description: Obsolete as of 23-Jan-2020. Use mnd1id 18804 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 38341	. 2 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp
3		opex 5428	. . . . 5 ⊢ ⟨𝐴, 𝐴⟩ ∈ V
4	3	rnsnop 6205	. . . 4 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴}
5	4	eqcomi 2770	. . 3 ⊢ {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}
6		eqid 2761	. . 3 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩})
7	5, 6	grpoidcl 30673	. 2 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴})
8		elsni 4596	. 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 1559   ∈ wcel 2141  Vcvv 3453  {csn 4579  ⟨cop 4585  ran crn 5644  ‘cfv 6515  GrpOpcgr 30648  GIdcgi 30649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-grpo 30652  df-gid 30653

This theorem is referenced by:  zrdivrng  38412