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Theorem nfvres 6416
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 5596 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
2 inss1 3994 . . . . 5 (𝐵 ∩ dom 𝐹) ⊆ 𝐵
31, 2eqsstri 3797 . . . 4 dom (𝐹𝐵) ⊆ 𝐵
43sseli 3759 . . 3 (𝐴 ∈ dom (𝐹𝐵) → 𝐴𝐵)
54con3i 151 . 2 𝐴𝐵 → ¬ 𝐴 ∈ dom (𝐹𝐵))
6 ndmfv 6409 . 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 1652  wcel 2155  cin 3733  c0 4081  dom cdm 5279  cres 5281  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-dm 5289  df-res 5291  df-iota 6033  df-fv 6078
This theorem is referenced by:  fveqres  6422  fvresval  32131  trpredlem1  32191  funpartfv  32517
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