Theorem xpinintabd 43857

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 43856	1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  {crab 3400   ∩ cin 3901  𝒫 cpw 4555  ∩ cint 4903   × cxp 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-in 3909  df-ss 3919  df-pw 4557  df-int 4904

This theorem is referenced by:  relintab  43860