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Theorem nfvres 6688
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 5853 . . . 4 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
2 inss1 4179 . . . 4 (𝐵 ∩ dom 𝐹) ⊆ 𝐵
31, 2eqsstri 3976 . . 3 dom (𝐹𝐵) ⊆ 𝐵
43sseli 3938 . 2 (𝐴 ∈ dom (𝐹𝐵) → 𝐴𝐵)
5 ndmfv 6682 . 2 𝐴 ∈ dom (𝐹𝐵) → ((𝐹𝐵)‘𝐴) = ∅)
64, 5nsyl5 162 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2114  cin 3907  c0 4265  dom cdm 5532  cres 5534  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-dm 5542  df-res 5544  df-iota 6293  df-fv 6342
This theorem is referenced by:  fveqres  6694  fvresval  33084  trpredlem1  33140  funpartfv  33480  setrec2lem1  45162
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