MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressnop0 Structured version   Unicode version

Theorem ressnop0 5990
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 4972 . . 3  |-  ( <. A ,  B >.  e.  ( C  X.  _V )  ->  A  e.  C
)
21con3i 135 . 2  |-  ( -.  A  e.  C  ->  -.  <. A ,  B >.  e.  ( C  X.  _V ) )
3 df-res 4952 . . . 4  |-  ( {
<. A ,  B >. }  |`  C )  =  ( { <. A ,  B >. }  i^i  ( C  X.  _V ) )
4 incom 3643 . . . 4  |-  ( {
<. A ,  B >. }  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  { <. A ,  B >. } )
53, 4eqtri 2480 . . 3  |-  ( {
<. A ,  B >. }  |`  C )  =  ( ( C  X.  _V )  i^i  { <. A ,  B >. } )
6 disjsn 4036 . . . 4  |-  ( ( ( C  X.  _V )  i^i  { <. A ,  B >. } )  =  (/) 
<->  -.  <. A ,  B >.  e.  ( C  X.  _V ) )
76biimpri 206 . . 3  |-  ( -. 
<. A ,  B >.  e.  ( C  X.  _V )  ->  ( ( C  X.  _V )  i^i 
{ <. A ,  B >. } )  =  (/) )
85, 7syl5eq 2504 . 2  |-  ( -. 
<. A ,  B >.  e.  ( C  X.  _V )  ->  ( { <. A ,  B >. }  |`  C )  =  (/) )
92, 8syl 16 1  |-  ( -.  A  e.  C  -> 
( { <. A ,  B >. }  |`  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070    i^i cin 3427   (/)c0 3737   {csn 3977   <.cop 3983    X. cxp 4938    |` cres 4942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-opab 4451  df-xp 4946  df-res 4952
This theorem is referenced by:  fvunsn  6011  fsnunres  6020  constr3pthlem1  23678  ex-res  23785  wfrlem14  27873
  Copyright terms: Public domain W3C validator