Theorem elid 6198

Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6197 after elrid 6048. (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 5813	. . . . 5 ⊢ Rel I
2		dfrel3 6197	. . . . 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 6048	. 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7		rexv 3482	. 2 ⊢ (∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
8	5, 6, 7	3bitri 300	1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 209   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  ⟨cop 4595   I cid 5555   ↾ cres 5663  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-res 5673

This theorem is referenced by: (None)