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Theorem ndmfvg 5452
Description: The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
Assertion
Ref Expression
ndmfvg  |-  ( ( A  e.  _V  /\  -.  A  e.  dom  F )  ->  ( F `  A )  =  (/) )

Proof of Theorem ndmfvg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 euex 2029 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
2 eldmg 4734 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  dom  F  <->  E. x  A F x ) )
31, 2syl5ibr 155 . . . 4  |-  ( A  e.  _V  ->  ( E! x  A F x  ->  A  e.  dom  F ) )
43con3d 620 . . 3  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  F  ->  -.  E! x  A F x ) )
5 tz6.12-2 5412 . . 3  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
64, 5syl6 33 . 2  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  F  ->  ( F `  A )  =  (/) ) )
76imp 123 1  |-  ( ( A  e.  _V  /\  -.  A  e.  dom  F )  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1999   _Vcvv 2686   (/)c0 3363   class class class wbr 3929   dom cdm 4539   ` cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-dm 4549  df-iota 5088  df-fv 5131
This theorem is referenced by:  ovprc  5806  sumnul  11200
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