Theorem inxpex 39239

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 5279	. 2 ⊢ (𝑅 ∈ 𝑊 → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
2		xpexg 7753	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝐴 × 𝐵) ∈ V)
3		inex2g 5280	. . 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 2145  Vcvv 3451   ∩ cin 3898   × cxp 5649

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-opab 5168  df-xp 5657  df-rel 5658

This theorem is used by:  xrninxpex  39317