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Theorem elid 6218
Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6217 after elrid 6063. (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
StepHypRef Expression
1 reli 5835 . . . . 5 Rel I
2 dfrel3 6217 . . . . 5 (Rel I ↔ ( I ↾ V) = I )
31, 2mpbi 230 . . . 4 ( I ↾ V) = I
43eqcomi 2745 . . 3 I = ( I ↾ V)
54eleq2i 2832 . 2 (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6 elrid 6063 . 2 (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7 rexv 3508 . 2 (∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
85, 6, 73bitri 297 1 (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  wex 1778  wcel 2107  wrex 3069  Vcvv 3479  cop 4631   I cid 5576  cres 5686  Rel wrel 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-res 5696
This theorem is referenced by: (None)
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