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Theorem elidinxpid 5910
Description: Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
elidinxpid (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elidinxpid
StepHypRef Expression
1 elidinxp 5909 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2 inidm 4198 . . 3 (𝐴𝐴) = 𝐴
32rexeqi 3419 . 2 (∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
41, 3bitri 276 1 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wcel 2107  wrex 3143  cin 3938  cop 4569   I cid 5457   × cxp 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560
This theorem is referenced by: (None)
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