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Theorem ndmovg 6245
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 6093 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eleq2 2503 . . . . . 6  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  <. A ,  B >.  e.  ( R  X.  S ) ) )
3 opelxp 4868 . . . . . 6  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3syl6bb 261 . . . . 5  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  ( A  e.  R  /\  B  e.  S ) ) )
54notbid 294 . . . 4  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  <. A ,  B >.  e.  dom  F  <->  -.  ( A  e.  R  /\  B  e.  S
) ) )
6 ndmfv 5713 . . . 4  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
75, 6syl6bir 229 . . 3  |-  ( dom 
F  =  ( R  X.  S )  -> 
( -.  ( A  e.  R  /\  B  e.  S )  ->  ( F `  <. A ,  B >. )  =  (/) ) )
87imp 429 . 2  |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> 
( F `  <. A ,  B >. )  =  (/) )
91, 8syl5eq 2486 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 369    = wceq 1369    e. wcel 1756   (/)c0 3636   <.cop 3882    X. cxp 4837   dom cdm 4839   ` cfv 5417  (class class class)co 6090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-xp 4845  df-dm 4849  df-iota 5380  df-fv 5425  df-ov 6093
This theorem is referenced by:  ndmov  6246  curry1val  6664  curry2val  6668  1div0  9994  repsundef  12408  cshnz  12428  mamufacex  18288  mavmulsolcl  18361  mavmul0g  18363  iscau2  20787  1div0apr  23660
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