Theorem xpdisjres 33241

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 5662	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		xpdisj1 6153	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ∅)
3	1, 2	eqtrid 2807	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Vcvv 3450   ∩ cin 4088  ∅c0 4277   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by:  padct  33359