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Theorem elid 6210
Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6209 after elrid 6055. (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 5832 . . . . 5 Rel I
2 dfrel3 6209 . . . . 5 (Rel I ↔ ( I ↾ V) = I )
31, 2mpbi 229 . . . 4 ( I ↾ V) = I
43eqcomi 2735 . . 3 I = ( I ↾ V)
54eleq2i 2818 . 2 (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6 elrid 6055 . 2 (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7 rexv 3490 . 2 (∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
85, 6, 73bitri 296 1 (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  wcel 2099  wrex 3060  Vcvv 3462  cop 4639   I cid 5579  cres 5684  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-res 5694
This theorem is referenced by: (None)
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