Theorem elid 6199

Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6198 after elrid 6049. (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 5814	. . . . 5 ⊢ Rel I
2		dfrel3 6198	. . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I )
3	1, 2	mpbi 233	. . . 4 ⊢ ( I ↾ V) = I
4	3	eqcomi 2778	. . 3 ⊢ I = ( I ↾ V)
5	4	eleq2i 2861	. 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6		elrid 6049	. 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7		rexv 3490	. 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 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  ⟨cop 4600   I cid 5556   ↾ cres 5664  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-res 5674

This theorem is referenced by: (None)