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Theorem xrninxp2 34636
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 34614 . 2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)}
2 3ancoma 1120 . . . 4 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢𝐴𝑢(𝑅𝑆)𝑥))
3 df-3an 1110 . . . 4 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥))
4 3anass 1117 . . . 4 ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
52, 3, 43bitr3i 293 . . 3 (((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
65opabbii 4908 . 2 {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
71, 6eqtri 2819 1 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 385  w3a 1108   = wceq 1653  wcel 2157  cin 3766   class class class wbr 4841  {copab 4903   × cxp 5308  cxrn 34459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-rel 5317  df-cnv 5318
This theorem is referenced by:  inxpxrn  34638
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