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Theorem xrninxp2 36256
Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)
Assertion
Ref Expression
xrninxp2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝐶,𝑥   𝑢,𝑅,𝑥   𝑢,𝑆,𝑥

Proof of Theorem xrninxp2
StepHypRef Expression
1 inxp2 36234 . 2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)}
2 an21 644 . . 3 (((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
32opabbii 5120 . 2 {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
41, 3eqtri 2765 1 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wcel 2110  cin 3865   class class class wbr 5053  {copab 5115   × cxp 5549  cxrn 36069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559
This theorem is referenced by:  inxpxrn  36258
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