Theorem inxpex 38362

Description: Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.)

Assertion

Ref	Expression
inxpex	⊢ ((𝑅 ∈ 𝑊 ∨ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)

Proof of Theorem inxpex

Step	Hyp	Ref	Expression
1		inex1g 5294	. 2 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
2		xpexg 7749	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝐴 × 𝐵) ∈ V)
3		inex2g 5295	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
4	2, 3	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
5	1, 4	jaoi 857	1 ⊢ ((𝑅 ∈ 𝑊 ∨ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∈ wcel 2109  Vcvv 3464   ∩ cin 3930   × cxp 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-opab 5187  df-xp 5665  df-rel 5666

This theorem is referenced by:  xrninxpex  38417