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Theorem nssdmovg 6430
Description: The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)
Assertion
Ref Expression
nssdmovg  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  ( A F B )  =  (/) )

Proof of Theorem nssdmovg
StepHypRef Expression
1 df-ov 6273 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ssel2 3484 . . . . 5  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  ->  <. A ,  B >.  e.  ( R  X.  S
) )
3 opelxp 5018 . . . . 5  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3sylib 196 . . . 4  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  -> 
( A  e.  R  /\  B  e.  S
) )
54stoic1a 1609 . . 3  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  -.  <. A ,  B >.  e. 
dom  F )
6 ndmfv 5872 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
75, 6syl 16 . 2  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  ( F `  <. A ,  B >. )  =  (/) )
81, 7syl5eq 2507 1  |-  ( ( dom  F  C_  ( 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 367    = wceq 1398    e. wcel 1823    C_ wss 3461   (/)c0 3783   <.cop 4022    X. cxp 4986   dom cdm 4988   ` cfv 5570  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  mpt2ndm0  6489
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