Theorem xpdisjres 32584

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 6155	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ∅)
3	1, 2	eqtrid 2783	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3464   ∩ cin 3930  ∅c0 4313   × cxp 5657   ↾ cres 5661

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-pr 5407

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-sn 4607  df-pr 4609  df-op 4613  df-opab 5187  df-xp 5665  df-rel 5666  df-res 5671

This theorem is referenced by:  padct  32702