Theorem xrninxp2 36446

Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)

Assertion

Ref	Expression
xrninxp2	⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}

Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝐶,𝑥   𝑢,𝑅,𝑥   𝑢,𝑆,𝑥

Proof of Theorem xrninxp2

Step	Hyp	Ref	Expression
1		inxp2 36424	. 2 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)}
2		an21 640	. . 3 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
3	2	opabbii 5137	. 2 ⊢ {⟨𝑢, 𝑥⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
4	1, 3	eqtri 2766	1 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   class class class wbr 5070  {copab 5132   × cxp 5578   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588

This theorem is referenced by:  inxpxrn  36448