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Theorem ndmfvg 5335
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 1978 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
2 eldmg 4631 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  dom  F  <->  E. x  A F x ) )
31, 2syl5ibr 154 . . . 4  |-  ( A  e.  _V  ->  ( E! x  A F x  ->  A  e.  dom  F ) )
43con3d 596 . . 3  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  F  ->  -.  E! x  A F x ) )
5 tz6.12-2 5296 . . 3  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
64, 5syl6 33 . 2  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  F  ->  ( F `  A )  =  (/) ) )
76imp 122 1  |-  ( ( A  e.  _V  /\  -.  A  e.  dom  F )  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   E!weu 1948   _Vcvv 2619   (/)c0 3286   class class class wbr 3845   dom cdm 4438   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-dm 4448  df-iota 4980  df-fv 5023
This theorem is referenced by:  ovprc  5684  sumnul  10814
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