Theorem inxp2 38349

Description: Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)

Assertion

Ref	Expression
inxp2	⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem inxp2

Step	Hyp	Ref	Expression
1		relinxp 5777	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2		dfrel4v 6163	. . 3 ⊢ (Rel (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦})
3	1, 2	mpbi 230	. 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦}
4		brinxp2 5716	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
5	4	opabbii 5174	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}
6	3, 5	eqtri 2752	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   class class class wbr 5107  {copab 5169   × cxp 5636  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646

This theorem is referenced by:  xrninxp  38378  xrninxp2  38379