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Theorem nfvres 6947
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 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)

Proof of Theorem nfvres
StepHypRef Expression
1 dmres 6030 . . . 4 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
2 inss1 4237 . . . 4 (𝐵 ∩ dom 𝐹) ⊆ 𝐵
31, 2eqsstri 4030 . . 3 dom (𝐹𝐵) ⊆ 𝐵
43sseli 3979 . 2 (𝐴 ∈ dom (𝐹𝐵) → 𝐴𝐵)
5 ndmfv 6941 . 2 𝐴 ∈ dom (𝐹𝐵) → ((𝐹𝐵)‘𝐴) = ∅)
64, 5nsyl5 159 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  cin 3950  c0 4333  dom cdm 5685  cres 5687  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-res 5697  df-iota 6514  df-fv 6569
This theorem is referenced by:  fveqres  6953  fvresval  7378  funpartfv  35946  setrec2lem1  49212
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