Theorem disjresin 38742

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

Assertion

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

Proof of Theorem disjresin

Step	Hyp	Ref	Expression
1		reseq2 5960	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = (𝑅 ↾ ∅))
2		res0 5969	. 2 ⊢ (𝑅 ↾ ∅) = ∅
3	1, 2	eqtrdi 2813	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1560   ∩ cin 3903  ∅c0 4285   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-in 3911  df-nul 4286  df-opab 5163  df-xp 5653  df-res 5659

This theorem is referenced by:  disjresdisj  38743