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Theorem idinxpres 6067
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 6068) 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 6064 . . 3 (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
2 elrid 6066 . . 3 (𝑥 ∈ ( I ↾ (𝐴𝐵)) ↔ ∃𝑦 ∈ (𝐴𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
31, 2bitr4i 278 . 2 (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ 𝑥 ∈ ( I ↾ (𝐴𝐵)))
43eqriv 2732 1 ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wrex 3068  cin 3962  cop 4637   I cid 5582   × cxp 5687  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  idinxpresid  6068
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