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Theorem nssdmovg 6231
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 6086 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ssel2 3345 . . . . . 6  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  ->  <. A ,  B >.  e.  ( R  X.  S
) )
3 opelxp 4910 . . . . . 6  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  <->  ( A  e.  R  /\  B  e.  S ) )
42, 3sylib 190 . . . . 5  |-  ( ( dom  F  C_  ( R  X.  S )  /\  <. A ,  B >.  e. 
dom  F )  -> 
( A  e.  R  /\  B  e.  S
) )
54ex 425 . . . 4  |-  ( dom 
F  C_  ( R  X.  S )  ->  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  R  /\  B  e.  S ) ) )
65con3and 430 . . 3  |-  ( ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S
) )  ->  -.  <. A ,  B >.  e. 
dom  F )
7 ndmfv 5757 . . 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 2482 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 360    = wceq 1653    e. wcel 1726    C_ wss 3322   (/)c0 3630   <.cop 3819    X. cxp 4878   dom cdm 4880   ` cfv 5456  (class class class)co 6083
This theorem is referenced by:  mpt2ndm0  6475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-dm 4890  df-iota 5420  df-fv 5464  df-ov 6086
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