Theorem idinxpres 6046

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 6047) to cartesian product. (Revised by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idinxpres	⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵))

Proof of Theorem idinxpres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elidinxp 6043	. . 3 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
2		elrid 6045	. . 3 ⊢ (𝑥 ∈ ( I ↾ (𝐴 ∩ 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
3	1, 2	bitr4i 278	. 2 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ 𝑥 ∈ ( I ↾ (𝐴 ∩ 𝐵)))
4	3	eqriv 2728	1 ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵))

Syntax hints:   = wceq 1540   ∈ wcel 2105  ∃wrex 3069   ∩ cin 3947  ⟨cop 4634   I cid 5573   × cxp 5674   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-res 5688

This theorem is referenced by:  idinxpresid  6047