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Theorem ressnop0 6087
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 4872 . . 3  |-  ( <. A ,  B >.  e.  ( C  X.  _V )  ->  A  e.  C
)
21con3i 142 . 2  |-  ( -.  A  e.  C  ->  -.  <. A ,  B >.  e.  ( C  X.  _V ) )
3 df-res 4851 . . . 4  |-  ( {
<. A ,  B >. }  |`  C )  =  ( { <. A ,  B >. }  i^i  ( C  X.  _V ) )
4 incom 3616 . . . 4  |-  ( {
<. A ,  B >. }  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  { <. A ,  B >. } )
53, 4eqtri 2493 . . 3  |-  ( {
<. A ,  B >. }  |`  C )  =  ( ( C  X.  _V )  i^i  { <. A ,  B >. } )
6 disjsn 4023 . . . 4  |-  ( ( ( C  X.  _V )  i^i  { <. A ,  B >. } )  =  (/) 
<->  -.  <. A ,  B >.  e.  ( C  X.  _V ) )
76biimpri 211 . . 3  |-  ( -. 
<. A ,  B >.  e.  ( C  X.  _V )  ->  ( ( C  X.  _V )  i^i 
{ <. A ,  B >. } )  =  (/) )
85, 7syl5eq 2517 . 2  |-  ( -. 
<. A ,  B >.  e.  ( C  X.  _V )  ->  ( { <. A ,  B >. }  |`  C )  =  (/) )
92, 8syl 17 1  |-  ( -.  A  e.  C  -> 
( { <. A ,  B >. }  |`  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389   (/)c0 3722   {csn 3959   <.cop 3965    X. cxp 4837    |` cres 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845  df-res 4851
This theorem is referenced by:  fvunsn  6112  fsnunres  6121  wfrlem14  7067  constr3pthlem1  25462  ex-res  25970
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