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Theorem nfvres 5719
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 3521 . . . . 5  |-  ( B  i^i  dom  F )  C_  B
31, 2eqsstri 3338 . . . 4  |-  dom  ( F  |`  B )  C_  B
43sseli 3304 . . 3  |-  ( A  e.  dom  ( F  |`  B )  ->  A  e.  B )
54con3i 129 . 2  |-  ( -.  A  e.  B  ->  -.  A  e.  dom  ( F  |`  B ) )
6 ndmfv 5714 . 2  |-  ( -.  A  e.  dom  ( F  |`  B )  -> 
( ( F  |`  B ) `  A
)  =  (/) )
75, 6syl 16 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1721    i^i cin 3279   (/)c0 3588   dom cdm 4837    |` cres 4839   ` cfv 5413
This theorem is referenced by:  fveqres  5723  fvresval  25337  trpredlem1  25444  funpartfv  25698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-dm 4847  df-res 4849  df-iota 5377  df-fv 5421
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