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Theorem idinxpres 6036
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 6037) 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 6033 . . 3 (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
2 elrid 6035 . . 3 (𝑥 ∈ ( I ↾ (𝐴𝐵)) ↔ ∃𝑦 ∈ (𝐴𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
31, 2bitr4i 277 . 2 (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ 𝑥 ∈ ( I ↾ (𝐴𝐵)))
43eqriv 2728 1 ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wrex 3069  cin 3943  cop 4628   I cid 5566   × cxp 5667  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-res 5681
This theorem is referenced by:  idinxpresid  6037
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