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Theorem nfvres 5640
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 5058 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
2 inss1 3465 . . . . 5  |-  ( B  i^i  dom  F )  C_  B
31, 2eqsstri 3284 . . . 4  |-  dom  ( F  |`  B )  C_  B
43sseli 3252 . . 3  |-  ( A  e.  dom  ( F  |`  B )  ->  A  e.  B )
54con3i 127 . 2  |-  ( -.  A  e.  B  ->  -.  A  e.  dom  ( F  |`  B ) )
6 ndmfv 5635 . 2  |-  ( -.  A  e.  dom  ( F  |`  B )  -> 
( ( F  |`  B ) `  A
)  =  (/) )
75, 6syl 15 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1642    e. wcel 1710    i^i cin 3227   (/)c0 3531   dom cdm 4771    |` cres 4773   ` cfv 5337
This theorem is referenced by:  fveqres  5643  fvresval  24681  trpredlem1  24788  funpartfv  25042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-xp 4777  df-dm 4781  df-res 4783  df-iota 5301  df-fv 5345
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