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Theorem elidinxpid 6054
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 6053 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2 inidm 4220 . . 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 1534  wcel 2099  wrex 3060  cin 3946  cop 4639   I cid 5579   × cxp 5680
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
This theorem is referenced by: (None)
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