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Theorem ndmovg 6457
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 6298 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eleq2 2520 . . . . . 6  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  <. A ,  B >.  e.  ( R  X.  S ) ) )
3 opelxp 4867 . . . . . 6  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3syl6bb 265 . . . . 5  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  ( A  e.  R  /\  B  e.  S ) ) )
54notbid 296 . . . 4  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  <. A ,  B >.  e.  dom  F  <->  -.  ( A  e.  R  /\  B  e.  S
) ) )
6 ndmfv 5894 . . . 4  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
75, 6syl6bir 233 . . 3  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  ( A  e.  R  /\  B  e.  S )  ->  ( F `  <. A ,  B >. )  =  (/) ) )
87imp 431 . 2  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( F `  <. A ,  B >. )  =  (/) )
91, 8syl5eq 2499 1  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   (/)c0 3733   <.cop 3976    X. cxp 4835   dom cdm 4837   ` cfv 5585  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-dm 4847  df-iota 5549  df-fv 5593  df-ov 6298
This theorem is referenced by:  ndmov  6458  curry1val  6894  curry2val  6898  1div0  10278  repsundef  12881  cshnz  12901  mamufacex  19426  mavmulsolcl  19588  mavmul0g  19590  iscau2  22259  1div0apr  25917
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