Theorem nfvres 6860

Description: The value of a non-member of a restriction is the empty set. (An artifact of our function value definition.) (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 5960	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
2		inss1 4184	. . . 4 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵
3	1, 2	eqsstri 3976	. . 3 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵
4	3	sseli 3925	. 2 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵)
5		ndmfv 6854	. 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
6	4, 5	nsyl5 159	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111   ∩ cin 3896  ∅c0 4280  dom cdm 5614   ↾ cres 5616  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-dm 5624  df-res 5626  df-iota 6437  df-fv 6489

This theorem is referenced by:  fveqres  6866  fvresval  7292  funpartfv  35989  setrec2lem1  49804