Theorem inxpex 39474

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 5284	. 2 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
2		xpexg 7791	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝐴 × 𝐵) ∈ V)
3		inex2g 5285	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
4	2, 3	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
5	1, 4	jaoi 871	1 ⊢ ((𝑅 ∈ 𝑊 ∨ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092   × cxp 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-opab 5172  df-xp 5662  df-rel 5663

This theorem is used by:  xrninxpex  39552