Theorem disjresdisj 37102

Description: The intersection of restrictions to disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)

Assertion

Ref	Expression
disjresdisj	⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)

Proof of Theorem disjresdisj

Step	Hyp	Ref	Expression
1		resindi 5997	. 2 ⊢ (𝑅 ↾ (𝐴 ∩ 𝐵)) = ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵))
2		disjresin 37101	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
3	1, 2	eqtr3id 2786	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3947  ∅c0 4322   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683  df-res 5688

This theorem is referenced by:  disjresdif  37103