Theorem resdisj 5196

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		resres 5055	. 2 ⊢ ((𝐶 ↾ 𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴 ∩ 𝐵))
2		reseq2 5038	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ↾ (𝐴 ∩ 𝐵)) = (𝐶 ↾ ∅))
3		res0 5047	. . 3 ⊢ (𝐶 ↾ ∅) = ∅
4	2, 3	eqtrdi 2283	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ↾ (𝐴 ∩ 𝐵)) = ∅)
5	1, 4	eqtrid 2279	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1398   ∩ cin 3213  ∅c0 3512   ↾ cres 4756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-xp 4760  df-rel 4761  df-res 4766

This theorem is referenced by:  fvsnun1  5886