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Theorem nfvres 5909
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 5131 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
2 inss1 3643 . . . . 5  |-  ( B  i^i  dom  F )  C_  B
31, 2eqsstri 3448 . . . 4  |-  dom  ( F  |`  B )  C_  B
43sseli 3414 . . 3  |-  ( A  e.  dom  ( F  |`  B )  ->  A  e.  B )
54con3i 142 . 2  |-  ( -.  A  e.  B  ->  -.  A  e.  dom  ( F  |`  B ) )
6 ndmfv 5903 . 2  |-  ( -.  A  e.  dom  ( F  |`  B )  -> 
( ( F  |`  B ) `  A
)  =  (/) )
75, 6syl 17 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904    i^i cin 3389   (/)c0 3722   dom cdm 4839    |` cres 4841   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-dm 4849  df-res 4851  df-iota 5553  df-fv 5597
This theorem is referenced by:  fveqres  5913  fvresval  30479  trpredlem1  30539  funpartfv  30783
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