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Theorem ndmovg 6170
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 6024 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eleq2 2449 . . . . . 6  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  <. A ,  B >.  e.  ( R  X.  S ) ) )
3 opelxp 4849 . . . . . 6  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3syl6bb 253 . . . . 5  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  ( A  e.  R  /\  B  e.  S ) ) )
54notbid 286 . . . 4  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  <. A ,  B >.  e.  dom  F  <->  -.  ( A  e.  R  /\  B  e.  S
) ) )
6 ndmfv 5696 . . . 4  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
75, 6syl6bir 221 . . 3  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  ( A  e.  R  /\  B  e.  S )  ->  ( F `  <. A ,  B >. )  =  (/) ) )
87imp 419 . 2  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( F `  <. A ,  B >. )  =  (/) )
91, 8syl5eq 2432 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 359    = wceq 1649    e. wcel 1717   (/)c0 3572   <.cop 3761    X. cxp 4817   dom cdm 4819   ` cfv 5395  (class class class)co 6021
This theorem is referenced by:  ndmov  6171  curry1val  6379  curry2val  6383  1div0  9612  iscau2  19102  1div0apr  21611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-dm 4829  df-iota 5359  df-fv 5403  df-ov 6024
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