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Theorem ndmaovg 27372
Description: The value of an operation outside its domain, analogous to ndmovg 6090. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
ndmaovg  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> (( A F B))  =  _V )

Proof of Theorem ndmaovg
StepHypRef Expression
1 opelxp 4801 . . . . 5  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
2 eleq2 2419 . . . . . 6  |-  ( ( R  X.  S )  =  dom  F  -> 
( <. A ,  B >.  e.  ( R  X.  S )  <->  <. A ,  B >.  e.  dom  F
) )
32eqcoms 2361 . . . . 5  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  ( R  X.  S )  <->  <. A ,  B >.  e.  dom  F
) )
41, 3syl5bbr 250 . . . 4  |-  ( dom 
F  =  ( R  X.  S )  -> 
( ( A  e.  R  /\  B  e.  S )  <->  <. A ,  B >.  e.  dom  F
) )
54notbid 285 . . 3  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  ( A  e.  R  /\  B  e.  S )  <->  -.  <. A ,  B >.  e.  dom  F
) )
65biimpa 470 . 2  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  ->  -.  <. A ,  B >.  e.  dom  F )
7 ndmaov 27371 . 2  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
86, 7syl 15 1  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> (( A F B))  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719    X. cxp 4769   dom cdm 4771   ((caov 27296
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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  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-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-opab 4159  df-xp 4777  df-fv 5345  df-dfat 27297  df-afv 27298  df-aov 27299
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