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Theorem nssdmovg 6188
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 6043 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ssel2 3303 . . . . . 6  |-  ( ( dom  F  C_  ( 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, 3sylib 189 . . . . 5  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  -> 
( A  e.  R  /\  B  e.  S
) )
54ex 424 . . . 4  |-  ( dom 
F  C_  ( R  X.  S )  ->  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  R  /\  B  e.  S ) ) )
65con3and 429 . . 3  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  -.  <. A ,  B >.  e. 
dom  F )
7 ndmfv 5714 . . 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 2448 1  |-  ( ( dom  F  C_  ( 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 359    = wceq 1649    e. wcel 1721    C_ wss 3280   (/)c0 3588   <.cop 3777    X. cxp 4835   dom cdm 4837   ` cfv 5413  (class class class)co 6040
This theorem is referenced by:  mpt2ndm0  6432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-dm 4847  df-iota 5377  df-fv 5421  df-ov 6043
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