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Theorem nfvres 6262
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 5454 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
2 inss1 3866 . . . . 5 (𝐵 ∩ dom 𝐹) ⊆ 𝐵
31, 2eqsstri 3668 . . . 4 dom (𝐹𝐵) ⊆ 𝐵
43sseli 3632 . . 3 (𝐴 ∈ dom (𝐹𝐵) → 𝐴𝐵)
54con3i 150 . 2 𝐴𝐵 → ¬ 𝐴 ∈ dom (𝐹𝐵))
6 ndmfv 6256 . 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 1523  wcel 2030  cin 3606  c0 3948  dom cdm 5143  cres 5145  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-res 5155  df-iota 5889  df-fv 5934
This theorem is referenced by:  fveqres  6268  fvresval  31791  trpredlem1  31851  funpartfv  32177
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