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Theorem suppssrOLD 5939
Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppssr 6823 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 3439 . 2  |-  ( X  e.  ( A  \  W )  <->  ( X  e.  A  /\  -.  X  e.  W ) )
2 fvex 5802 . . . . . 6  |-  ( F `
 X )  e. 
_V
3 eldifsn 4101 . . . . . 6  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( ( F `  X )  e.  _V  /\  ( F `
 X )  =/= 
Z ) )
42, 3mpbiran 909 . . . . 5  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( F `  X )  =/=  Z
)
5 suppssrOLD.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
6 ffn 5660 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
7 elpreima 5925 . . . . . . . 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 3456 . . . . . . 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 2666 . . 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 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071    \ cdif 3426    C_ wss 3429   {csn 3978   `'ccnv 4940   "cima 4944    Fn wfn 5514   -->wf 5515   ` cfv 5519
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 4514  ax-nul 4522  ax-pr 4632
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-eu 2264  df-mo 2265  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 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527
This theorem is referenced by:  cantnfp1lem1OLD  8016  cantnfp1lem3OLD  8018  cantnflem1dOLD  8023  cantnflem1OLD  8024  cnfcom2lemOLD  8046  gsumval3OLD  16495  gsumcllemOLD  16500  gsumzaddlemOLD  16523  gsumzsplitOLD  16532  gsumzmhmOLD  16545  gsumzinvOLD  16557  gsumsubOLD  16562  gsumptOLD  16569  gsum2dOLD  16578  dprdfinvOLD  16630  dprdfaddOLD  16631  dmdprdsplitlemOLD  16649  dpjidclOLD  16678  gsumdixpOLD  16815  lcomfsupOLD  17099  psrbaglesuppOLD  17551  psrbagaddclOLD  17554  psrbaglefiOLD  17557  mplsubglemOLD  17628  mpllsslemOLD  17629  mplsubrglemOLD  17634  mplcoe2OLD  17666  mplbas2OLD  17668  evlslem4OLD  17706  frlmsslspOLD  18342
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