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Theorem elid 6155
Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6154 after elrid 6003. (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 5786 . . . . 5 Rel I
2 dfrel3 6154 . . . . 5 (Rel I ↔ ( I ↾ V) = I )
31, 2mpbi 229 . . . 4 ( I ↾ V) = I
43eqcomi 2742 . . 3 I = ( I ↾ V)
54eleq2i 2826 . 2 (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6 elrid 6003 . 2 (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7 rexv 3472 . 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 205   = wceq 1542  wex 1782  wcel 2107  wrex 3070  Vcvv 3447  cop 4596   I cid 5534  cres 5639  Rel wrel 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-res 5649
This theorem is referenced by: (None)
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