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Theorem ressnop0 5845
Description: If  A is not in  C, then the restriction of a singleton of  <. A ,  B >. to  C is null. (Contributed by Scott Fenton, 15-Apr-2011.)
Assertion
Ref Expression
ressnop0  |-  ( -.  A  e.  C  -> 
( { <. A ,  B >. }  |`  C )  =  (/) )

Proof of Theorem ressnop0
StepHypRef Expression
1 opelxp1 4844 . . 3  |-  ( <. A ,  B >.  e.  ( C  X.  _V )  ->  A  e.  C
)
21con3i 129 . 2  |-  ( -.  A  e.  C  ->  -.  <. A ,  B >.  e.  ( C  X.  _V ) )
3 df-res 4823 . . . 4  |-  ( {
<. A ,  B >. }  |`  C )  =  ( { <. A ,  B >. }  i^i  ( C  X.  _V ) )
4 incom 3469 . . . 4  |-  ( {
<. A ,  B >. }  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  { <. A ,  B >. } )
53, 4eqtri 2400 . . 3  |-  ( {
<. A ,  B >. }  |`  C )  =  ( ( C  X.  _V )  i^i  { <. A ,  B >. } )
6 disjsn 3804 . . . 4  |-  ( ( ( C  X.  _V )  i^i  { <. A ,  B >. } )  =  (/) 
<->  -.  <. A ,  B >.  e.  ( C  X.  _V ) )
76biimpri 198 . . 3  |-  ( -. 
<. A ,  B >.  e.  ( C  X.  _V )  ->  ( ( C  X.  _V )  i^i 
{ <. A ,  B >. } )  =  (/) )
85, 7syl5eq 2424 . 2  |-  ( -. 
<. A ,  B >.  e.  ( C  X.  _V )  ->  ( { <. A ,  B >. }  |`  C )  =  (/) )
92, 8syl 16 1  |-  ( -.  A  e.  C  -> 
( { <. A ,  B >. }  |`  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2892    i^i cin 3255   (/)c0 3564   {csn 3750   <.cop 3753    X. cxp 4809    |` cres 4813
This theorem is referenced by:  fvunsn  5857  fsnunres  5866  constr3pthlem1  21483  ex-res  21590  wfrlem14  25286
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  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-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-opab 4201  df-xp 4817  df-res 4823
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