Theorem xrninxpex 38784

Description: Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.)

Assertion

Ref	Expression
xrninxpex	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)

Proof of Theorem xrninxpex

Step	Hyp	Ref	Expression
1		xpexg 7693	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
2		inxpex 38706	. . . 4 ⊢ (((𝑅 ⋉ 𝑆) ∈ V ∨ (𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
3	2	olcs 882	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
4	1, 3	sylan2 599	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
5	4	3impb 1120	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  Vcvv 3431   ∩ cin 3882   × cxp 5616   ⋉ cxrn 38541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-opab 5135  df-xp 5624  df-rel 5625

This theorem is referenced by:  xrnresex  38796