Theorem resdisj 6150

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 5956	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ↾ (𝐴 ∩ 𝐵)) = (𝐶 ↾ ∅))
2		resres 5974	. 2 ⊢ ((𝐶 ↾ 𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴 ∩ 𝐵))
3		res0 5965	. . 3 ⊢ (𝐶 ↾ ∅) = ∅
4	3	eqcomi 2770	. 2 ⊢ ∅ = (𝐶 ↾ ∅)
5	1, 2, 4	3eqtr4g 2821	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1559   ∩ cin 3901  ∅c0 4283   ↾ cres 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-xp 5649  df-rel 5650  df-res 5655

This theorem is referenced by:  fvsnun1  7161