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Theorem nfvres 5715
Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
Assertion
Ref Expression
nfvres  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )

Proof of Theorem nfvres
StepHypRef Expression
1 dmres 5126 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
2 inss1 3565 . . . . 5  |-  ( B  i^i  dom  F )  C_  B
31, 2eqsstri 3381 . . . 4  |-  dom  ( F  |`  B )  C_  B
43sseli 3347 . . 3  |-  ( A  e.  dom  ( F  |`  B )  ->  A  e.  B )
54con3i 135 . 2  |-  ( -.  A  e.  B  ->  -.  A  e.  dom  ( F  |`  B ) )
6 ndmfv 5709 . 2  |-  ( -.  A  e.  dom  ( F  |`  B )  -> 
( ( F  |`  B ) `  A
)  =  (/) )
75, 6syl 16 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756    i^i cin 3322   (/)c0 3632   dom cdm 4835    |` cres 4837   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-dm 4845  df-res 4847  df-iota 5376  df-fv 5421
This theorem is referenced by:  fveqres  5719  fvresval  27529  trpredlem1  27642  funpartfv  27927
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