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Theorem suppssrOLD 6013
Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppssr 6928 as of 28-May-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
suppssrOLD.f  |-  ( ph  ->  F : A --> B )
suppssrOLD.n  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
Assertion
Ref Expression
suppssrOLD  |-  ( (
ph  /\  X  e.  ( A  \  W ) )  ->  ( F `  X )  =  Z )

Proof of Theorem suppssrOLD
StepHypRef Expression
1 eldif 3486 . 2  |-  ( X  e.  ( A  \  W )  <->  ( X  e.  A  /\  -.  X  e.  W ) )
2 fvex 5874 . . . . . 6  |-  ( F `
 X )  e. 
_V
3 eldifsn 4152 . . . . . 6  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( ( F `  X )  e.  _V  /\  ( F `
 X )  =/= 
Z ) )
42, 3mpbiran 916 . . . . 5  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( F `  X )  =/=  Z
)
5 suppssrOLD.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
6 ffn 5729 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
7 elpreima 5999 . . . . . . . 8  |-  ( F  Fn  A  ->  ( X  e.  ( `' F " ( _V  \  { Z } ) )  <-> 
( X  e.  A  /\  ( F `  X
)  e.  ( _V 
\  { Z }
) ) ) )
85, 6, 73syl 20 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( X  e.  A  /\  ( F `  X )  e.  ( _V  \  { Z } ) ) ) )
9 suppssrOLD.n . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
109sseld 3503 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  { Z }
) )  ->  X  e.  W ) )
118, 10sylbird 235 . . . . . 6  |-  ( ph  ->  ( ( X  e.  A  /\  ( F `
 X )  e.  ( _V  \  { Z } ) )  ->  X  e.  W )
)
1211expdimp 437 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
)  e.  ( _V 
\  { Z }
)  ->  X  e.  W ) )
134, 12syl5bir 218 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
)  =/=  Z  ->  X  e.  W )
)
1413necon1bd 2685 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( -.  X  e.  W  ->  ( F `  X
)  =  Z ) )
1514impr 619 . 2  |-  ( (
ph  /\  ( X  e.  A  /\  -.  X  e.  W ) )  -> 
( F `  X
)  =  Z )
161, 15sylan2b 475 1  |-  ( (
ph  /\  X  e.  ( A  \  W ) )  ->  ( F `  X )  =  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {csn 4027   `'ccnv 4998   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594
This theorem is referenced by:  cantnfp1lem1OLD  8119  cantnfp1lem3OLD  8121  cantnflem1dOLD  8126  cantnflem1OLD  8127  cnfcom2lemOLD  8149  gsumval3OLD  16699  gsumcllemOLD  16704  gsumzaddlemOLD  16727  gsumzsplitOLD  16736  gsumzmhmOLD  16749  gsumzinvOLD  16761  gsumsubOLD  16766  gsumptOLD  16780  gsum2dOLD  16791  dprdfinvOLD  16856  dprdfaddOLD  16857  dmdprdsplitlemOLD  16875  dpjidclOLD  16904  gsumdixpOLD  17041  lcomfsupOLD  17332  psrbaglesuppOLD  17789  psrbagaddclOLD  17792  psrbaglefiOLD  17795  mplsubglemOLD  17866  mpllsslemOLD  17867  mplsubrglemOLD  17872  mplcoe2OLD  17904  mplbas2OLD  17906  evlslem4OLD  17944  frlmsslspOLD  18597
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