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Theorem ndmaovg 28038
Description: The value of an operation outside its domain, analogous to ndmovg 6233. (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 4911 . . . . 5  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
2 eleq2 2499 . . . . . 6  |-  ( ( R  X.  S )  =  dom  F  -> 
( <. A ,  B >.  e.  ( R  X.  S )  <->  <. A ,  B >.  e.  dom  F
) )
32eqcoms 2441 . . . . 5  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  ( R  X.  S )  <->  <. A ,  B >.  e.  dom  F
) )
41, 3syl5bbr 252 . . . 4  |-  ( dom 
F  =  ( R  X.  S )  -> 
( ( A  e.  R  /\  B  e.  S )  <->  <. A ,  B >.  e.  dom  F
) )
54notbid 287 . . 3  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  ( A  e.  R  /\  B  e.  S )  <->  -.  <. A ,  B >.  e.  dom  F
) )
65biimpa 472 . 2  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  ->  -.  <. A ,  B >.  e.  dom  F )
7 ndmaov 28037 . 2  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
86, 7syl 16 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    X. cxp 4879   dom cdm 4881   ((caov 27963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887  df-fv 5465  df-dfat 27964  df-afv 27965  df-aov 27966
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