Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xpinintabd Structured version   Visualization version   GIF version

Theorem xpinintabd 41859
Description: Value of the intersection of Cartesian 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 41858 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 397   = wceq 1542  wex 1782  {cab 2714  {crab 3408  cin 3910  𝒫 cpw 4561   cint 4908   × cxp 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-in 3918  df-ss 3928  df-pw 4563  df-int 4909
This theorem is referenced by:  relintab  41862
  Copyright terms: Public domain W3C validator