Theorem disjresin 38954

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 5975	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = (𝑅 ↾ ∅))
2		res0 5984	. 2 ⊢ (𝑅 ↾ ∅) = ∅
3	1, 2	eqtrdi 2816	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 3905  ∅c0 4286   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-in 3913  df-nul 4287  df-opab 5176  df-xp 5669  df-res 5675

This theorem is used by:  disjresdisj  38955