Theorem inxp2 38395

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 5749	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2		dfrel4v 6132	. . 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 5689	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
5	4	opabbii 5153	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}
6	3, 5	eqtri 2754	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∩ cin 3896   class class class wbr 5086  {copab 5148   × cxp 5609  Rel wrel 5616

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619

This theorem is referenced by:  xrninxp  38424  xrninxp2  38425