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Theorem idinxpres 6064
Description: The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 6065) to cartesian product. (Revised by BJ, 23-Dec-2023.)
Assertion
Ref Expression
idinxpres ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴𝐵))

Proof of Theorem idinxpres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elidinxp 6061 . . 3 (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
2 elrid 6063 . . 3 (𝑥 ∈ ( I ↾ (𝐴𝐵)) ↔ ∃𝑦 ∈ (𝐴𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
31, 2bitr4i 278 . 2 (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ 𝑥 ∈ ( I ↾ (𝐴𝐵)))
43eqriv 2733 1 ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  wrex 3069  cin 3949  cop 4631   I cid 5576   × cxp 5682  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-res 5696
This theorem is referenced by:  idinxpresid  6065
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