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Theorem nfvres 5804
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 5206 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
2 inss1 3632 . . . . 5  |-  ( B  i^i  dom  F )  C_  B
31, 2eqsstri 3447 . . . 4  |-  dom  ( F  |`  B )  C_  B
43sseli 3413 . . 3  |-  ( A  e.  dom  ( F  |`  B )  ->  A  e.  B )
54con3i 135 . 2  |-  ( -.  A  e.  B  ->  -.  A  e.  dom  ( F  |`  B ) )
6 ndmfv 5798 . 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 1399    e. wcel 1826    i^i cin 3388   (/)c0 3711   dom cdm 4913    |` cres 4915   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-xp 4919  df-dm 4923  df-res 4925  df-iota 5460  df-fv 5504
This theorem is referenced by:  fveqres  5808  fvresval  29363  trpredlem1  29475  funpartfv  29748
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