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Theorem nssdmovg 6347
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 6195 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ssel2 3451 . . . . . 6  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  ->  <. A ,  B >.  e.  ( R  X.  S
) )
3 opelxp 4969 . . . . . 6  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3sylib 196 . . . . 5  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  -> 
( A  e.  R  /\  B  e.  S
) )
54ex 434 . . . 4  |-  ( dom 
F  C_  ( R  X.  S )  ->  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  R  /\  B  e.  S ) ) )
65con3dimp 441 . . 3  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  -.  <. A ,  B >.  e. 
dom  F )
7 ndmfv 5815 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
86, 7syl 16 . 2  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  ( F `  <. A ,  B >. )  =  (/) )
91, 8syl5eq 2504 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 369    = wceq 1370    e. wcel 1758    C_ wss 3428   (/)c0 3737   <.cop 3983    X. cxp 4938   dom cdm 4940   ` cfv 5518  (class class class)co 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-xp 4946  df-dm 4950  df-iota 5481  df-fv 5526  df-ov 6195
This theorem is referenced by:  mpt2ndm0  6841
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