Theorem xpdisjres 32880

Description: Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Assertion

Ref	Expression
xpdisjres	⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)

Proof of Theorem xpdisjres

Step	Hyp	Ref	Expression
1		df-res 5671	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		xpdisj1 6156	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ∅)
3	1, 2	eqtrid 2816	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1567  Vcvv 3463   ∩ cin 3912  ∅c0 4294   × cxp 5657   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5175  df-xp 5665  df-rel 5666  df-res 5671

This theorem is referenced by:  padct  33000