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Theorem nssdmovg 6161
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 6016 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ssel2 3279 . . . . . 6  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  ->  <. A ,  B >.  e.  ( R  X.  S
) )
3 opelxp 4841 . . . . . 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 5688 . . 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 2424 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 1717    C_ wss 3256   (/)c0 3564   <.cop 3753    X. cxp 4809   dom cdm 4811   ` cfv 5387  (class class class)co 6013
This theorem is referenced by:  mpt2ndm0  6402
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-dm 4821  df-iota 5351  df-fv 5395  df-ov 6016
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