Theorem resdisj 6162

Description: A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
resdisj	⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅)

Proof of Theorem resdisj

Step	Hyp	Ref	Expression
1		reseq2 5965	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ↾ (𝐴 ∩ 𝐵)) = (𝐶 ↾ ∅))
2		resres 5983	. 2 ⊢ ((𝐶 ↾ 𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴 ∩ 𝐵))
3		res0 5974	. . 3 ⊢ (𝐶 ↾ ∅) = ∅
4	3	eqcomi 2769	. 2 ⊢ ∅ = (𝐶 ↾ ∅)
5	1, 2, 4	3eqtr4g 2820	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∩ cin 4088  ∅c0 4277   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by:  fvsnun1  7189