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Theorem suppssrOLD 6022
Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppssr 6949 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 3481 . 2  |-  ( X  e.  ( A  \  W )  <->  ( X  e.  A  /\  -.  X  e.  W ) )
2 fvex 5882 . . . . . 6  |-  ( F `
 X )  e. 
_V
3 eldifsn 4157 . . . . . 6  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( ( F `  X )  e.  _V  /\  ( F `
 X )  =/= 
Z ) )
42, 3mpbiran 918 . . . . 5  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( F `  X )  =/=  Z
)
5 suppssrOLD.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
6 ffn 5737 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
7 elpreima 6008 . . . . . . . 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 3498 . . . . . . 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 2675 . . 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468    C_ wss 3471   {csn 4032   `'ccnv 5007   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  cantnfp1lem1OLD  8140  cantnfp1lem3OLD  8142  cantnflem1dOLD  8147  cantnflem1OLD  8148  cnfcom2lemOLD  8170  gsumval3OLD  17035  gsumcllemOLD  17040  gsumzaddlemOLD  17063  gsumzsplitOLD  17072  gsumzmhmOLD  17085  gsumzinvOLD  17097  gsumsubOLD  17102  gsumptOLD  17116  gsum2dOLD  17127  dprdfinvOLD  17193  dprdfaddOLD  17194  dmdprdsplitlemOLD  17212  dpjidclOLD  17241  gsumdixpOLD  17384  lcomfsupOLD  17676  psrbaglesuppOLD  18145  psrbagaddclOLD  18148  psrbaglefiOLD  18151  mplsubglemOLD  18222  mpllsslemOLD  18223  mplsubrglemOLD  18228  mplcoe2OLD  18260  mplbas2OLD  18262  evlslem4OLD  18300  frlmsslspOLD  18957
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