Theorem elid 6199

Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6198 after elrid 6043. (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 5808	. . . . 5 ⊢ Rel I
2		dfrel3 6198	. . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I )
3	1, 2	mpbi 233	. . . 4 ⊢ ( I ↾ V) = I
4	3	eqcomi 2772	. . 3 ⊢ I = ( I ↾ V)
5	4	eleq2i 2855	. 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6		elrid 6043	. 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7		rexv 3480	. 2 ⊢ (∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
8	5, 6, 7	3bitri 300	1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∃wrex 3089  Vcvv 3453  ⟨cop 4837   I cid 5548   ↾ cres 5656  Rel wrel 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-res 5666

This theorem is used by: (None)