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Theorem elidinxpid 6043
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 6042 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2 inidm 4213 . . 3 (𝐴𝐴) = 𝐴
32rexeqi 3314 . 2 (∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
41, 3bitri 274 1 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wrex 3060  cin 3939  cop 4630   I cid 5569   × cxp 5670
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679
This theorem is referenced by: (None)
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