Theorem idinxpres 6040

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 6041) 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 6037	. . 3 ⊢ (𝑥 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑦 ∈ (𝐴 ∩ 𝐵)𝑥 = ⟨𝑦, 𝑦⟩)
2		elrid 6039	. . 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 2723	1 ⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵))

Syntax hints:   = wceq 1533   ∈ wcel 2098  ∃wrex 3064   ∩ cin 3942  ⟨cop 4629   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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-res 5681

This theorem is referenced by:  idinxpresid  6041