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Theorem xpinintabd 36905
 Description: Value of the intersection of cross-product with the intersection of a non-empty class. (Contributed by RP, 12-Aug-2020.)
Hypothesis
Ref Expression
xpinintabd.x (𝜑 → ∃𝑥𝜓)
Assertion
Ref Expression
xpinintabd (𝜑 → ((𝐴 × 𝐵) ∩ {𝑥𝜓}) = {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)})
Distinct variable groups:   𝜓,𝑤   𝑥,𝑤,𝐴   𝑤,𝐵,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑤)   𝜓(𝑥)

Proof of Theorem xpinintabd
StepHypRef Expression
1 xpinintabd.x . 2 (𝜑 → ∃𝑥𝜓)
21inintabd 36904 1 (𝜑 → ((𝐴 × 𝐵) ∩ {𝑥𝜓}) = {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695  {cab 2596  {crab 2900   ∩ cin 3539  𝒫 cpw 4108  ∩ cint 4410   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-int 4411 This theorem is referenced by:  relintab  36908
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