Theorem disjresdisj 38953

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 5996	. 2 ⊢ (𝑅 ↾ (𝐴 ∩ 𝐵)) = ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵))
2		disjresin 38952	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
3	1, 2	eqtr3id 2814	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 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  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-xp 5669  df-rel 5670  df-res 5675

This theorem is used by:  disjresdif  38954