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Theorem elrid 5900
 Description: Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
elrid (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem elrid
StepHypRef Expression
1 df-res 5554 . . 3 ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
21eleq2i 2907 . 2 (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V)))
3 elidinxp 5898 . 2 (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩)
4 inv1 4331 . . 3 (𝑋 ∩ V) = 𝑋
54rexeqi 3401 . 2 (∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
62, 3, 53bitri 300 1 (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ∃wrex 3134  Vcvv 3480   ∩ cin 3918  ⟨cop 4556   I cid 5446   × cxp 5540   ↾ cres 5544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-res 5554 This theorem is referenced by:  idinxpres  5901  idrefALT  5960  elid  6043
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