Theorem inxp2 38472

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 5760	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2		dfrel4v 6145	. . 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 5699	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
5	4	opabbii 5162	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}
6	3, 5	eqtri 2756	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∩ cin 3897   class class class wbr 5095  {copab 5157   × cxp 5619  Rel wrel 5626

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629

This theorem is referenced by:  xrninxp  38512  xrninxp2  38513