Theorem disjresin 38282

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 5922	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = (𝑅 ↾ ∅))
2		res0 5931	. 2 ⊢ (𝑅 ↾ ∅) = ∅
3	1, 2	eqtrdi 2782	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3896  ∅c0 4280   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-in 3904  df-nul 4281  df-opab 5152  df-xp 5620  df-res 5626

This theorem is referenced by:  disjresdisj  38283