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Theorem nfvres 6134
 Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
Assertion
Ref Expression
nfvres 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)

Proof of Theorem nfvres
StepHypRef Expression
1 dmres 5339 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
2 inss1 3795 . . . . 5 (𝐵 ∩ dom 𝐹) ⊆ 𝐵
31, 2eqsstri 3598 . . . 4 dom (𝐹𝐵) ⊆ 𝐵
43sseli 3564 . . 3 (𝐴 ∈ dom (𝐹𝐵) → 𝐴𝐵)
54con3i 149 . 2 𝐴𝐵 → ¬ 𝐴 ∈ dom (𝐹𝐵))
6 ndmfv 6128 . 2 𝐴 ∈ dom (𝐹𝐵) → ((𝐹𝐵)‘𝐴) = ∅)
75, 6syl 17 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ∅c0 3874  dom cdm 5038   ↾ cres 5040  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050  df-iota 5768  df-fv 5812 This theorem is referenced by:  fveqres  6140  fvresval  30911  trpredlem1  30971  funpartfv  31222
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