Theorem disjresdisj 38222

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 6016	. 2 ⊢ (𝑅 ↾ (𝐴 ∩ 𝐵)) = ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵))
2		disjresin 38221	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
3	1, 2	eqtr3id 2789	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3962  ∅c0 4339   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-res 5701

This theorem is referenced by:  disjresdif  38223