Theorem nfvres 6262

Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Assertion

Ref	Expression
nfvres	⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Proof of Theorem nfvres

Step	Hyp	Ref	Expression
1		dmres 5454	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
2		inss1 3866	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵
3	1, 2	eqsstri 3668	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵
4	3	sseli 3632	. . 3 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵)
5	4	con3i 150	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵))
6		ndmfv 6256	. 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
7	5, 6	syl 17	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030   ∩ cin 3606  ∅c0 3948  dom cdm 5143   ↾ cres 5145  ‘cfv 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-res 5155  df-iota 5889  df-fv 5934

This theorem is referenced by:  fveqres  6268  fvresval  31791  trpredlem1  31851  funpartfv  32177