Theorem xpinintabd 43075

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) ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {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

Step	Hyp	Ref	Expression
1		xpinintabd.x	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2	1	inintabd 43074	1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773  {cab 2702  {crab 3419   ∩ cin 3938  𝒫 cpw 4598  ∩ cint 4944   × cxp 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-in 3946  df-ss 3956  df-pw 4600  df-int 4945

This theorem is referenced by:  relintab  43078