Theorem elid 6102

Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6101 after elrid 5953. (Contributed by BJ, 28-Aug-2022.)

Assertion

Ref	Expression
elid	⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

Distinct variable group:   𝑥,𝐴

Proof of Theorem elid

Step	Hyp	Ref	Expression
1		reli 5736	. . . . 5 ⊢ Rel I
2		dfrel3 6101	. . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I )
3	1, 2	mpbi 229	. . . 4 ⊢ ( I ↾ V) = I
4	3	eqcomi 2747	. . 3 ⊢ I = ( I ↾ V)
5	4	eleq2i 2830	. 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6		elrid 5953	. 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7		rexv 3457	. 2 ⊢ (∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
8	5, 6, 7	3bitri 297	1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 205   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432  ⟨cop 4567   I cid 5488   ↾ cres 5591  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-res 5601

This theorem is referenced by: (None)