Theorem inxp2 38909

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 5799	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2		dfrel4v 6187	. . 3 ⊢ (Rel (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦})
3	1, 2	mpbi 233	. 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦}
4		brinxp2 5737	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
5	4	opabbii 5179	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}
6	3, 5	eqtri 2792	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∩ cin 3912   class class class wbr 5110  {copab 5174   × cxp 5657  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667

This theorem is referenced by:  xrninxp  38949  xrninxp2  38950