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Theorem ndmovg 6087
Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
Assertion
Ref Expression
ndmovg  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  =  (/) )

Proof of Theorem ndmovg
StepHypRef Expression
1 df-ov 5945 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eleq2 2419 . . . . . 6  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  <. A ,  B >.  e.  ( R  X.  S ) ) )
3 opelxp 4798 . . . . . 6  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3syl6bb 252 . . . . 5  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  ( A  e.  R  /\  B  e.  S ) ) )
54notbid 285 . . . 4  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  <. A ,  B >.  e.  dom  F  <->  -.  ( A  e.  R  /\  B  e.  S
) ) )
6 ndmfv 5632 . . . 4  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
75, 6syl6bir 220 . . 3  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  ( A  e.  R  /\  B  e.  S )  ->  ( F `  <. A ,  B >. )  =  (/) ) )
87imp 418 . 2  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( F `  <. A ,  B >. )  =  (/) )
91, 8syl5eq 2402 1  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   (/)c0 3531   <.cop 3719    X. cxp 4766   dom cdm 4768   ` cfv 5334  (class class class)co 5942
This theorem is referenced by:  ndmov  6088  curry1val  6295  curry2val  6299  1div0  9512  iscau2  18801  1div0apr  20947
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 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
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 3907  df-br 4103  df-opab 4157  df-xp 4774  df-dm 4778  df-iota 5298  df-fv 5342  df-ov 5945
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