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Theorem xpinintabd 40075
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 40074 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 398   = wceq 1537  wex 1780  {cab 2798  {crab 3129  cin 3912  𝒫 cpw 4515   cint 4852   × cxp 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3475  df-in 3920  df-ss 3930  df-pw 4517  df-int 4853
This theorem is referenced by:  relintab  40078
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