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Theorem ndmovg 6003
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 5861 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eleq2 2344 . . . . . 6  |-  ( dom 
F  =  ( R  X.  S )  -> 
( <. A ,  B >.  e.  dom  F  <->  <. A ,  B >.  e.  ( R  X.  S ) ) )
3 opelxp 4719 . . . . . 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 5552 . . . 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 2327 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 1623    e. wcel 1684   (/)c0 3455   <.cop 3643    X. cxp 4687   dom cdm 4689   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  ndmov  6004  curry1val  6211  curry2val  6215  1div0  9425  iscau2  18703  1div0apr  20841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861
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