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Theorem xpinintabd 38412
Description: Value of the intersection of cross-product with the intersection of a nonempty 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 38411 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 382   = wceq 1631  wex 1852  {cab 2757  {crab 3065  cin 3722  𝒫 cpw 4298   cint 4612   × cxp 5248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-in 3730  df-ss 3737  df-pw 4300  df-int 4613
This theorem is referenced by:  relintab  38415
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