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Theorem elid 6221
Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6220 after elrid 6066. (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 5839 . . . . 5 Rel I
2 dfrel3 6220 . . . . 5 (Rel I ↔ ( I ↾ V) = I )
31, 2mpbi 230 . . . 4 ( I ↾ V) = I
43eqcomi 2744 . . 3 I = ( I ↾ V)
54eleq2i 2831 . 2 (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6 elrid 6066 . 2 (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7 rexv 3507 . 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 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  cop 4637   I cid 5582  cres 5691  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-res 5701
This theorem is referenced by: (None)
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