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Theorem suppssr 5856
Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
suppssr.f  |-  ( ph  ->  F : A --> B )
suppssr.n  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
Assertion
Ref Expression
suppssr  |-  ( (
ph  /\  X  e.  ( A  \  W ) )  ->  ( F `  X )  =  Z )

Proof of Theorem suppssr
StepHypRef Expression
1 eldif 3322 . 2  |-  ( X  e.  ( A  \  W )  <->  ( X  e.  A  /\  -.  X  e.  W ) )
2 fvex 5734 . . . . . 6  |-  ( F `
 X )  e. 
_V
3 eldifsn 3919 . . . . . 6  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( ( F `  X )  e.  _V  /\  ( F `
 X )  =/= 
Z ) )
42, 3mpbiran 885 . . . . 5  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( F `  X )  =/=  Z
)
5 suppssr.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
6 ffn 5583 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
7 elpreima 5842 . . . . . . . 8  |-  ( F  Fn  A  ->  ( X  e.  ( `' F " ( _V  \  { Z } ) )  <-> 
( X  e.  A  /\  ( F `  X
)  e.  ( _V 
\  { Z }
) ) ) )
85, 6, 73syl 19 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( X  e.  A  /\  ( F `  X )  e.  ( _V  \  { Z } ) ) ) )
9 suppssr.n . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
109sseld 3339 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  { Z }
) )  ->  X  e.  W ) )
118, 10sylbird 227 . . . . . 6  |-  ( ph  ->  ( ( X  e.  A  /\  ( F `
 X )  e.  ( _V  \  { Z } ) )  ->  X  e.  W )
)
1211expdimp 427 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
)  e.  ( _V 
\  { Z }
)  ->  X  e.  W ) )
134, 12syl5bir 210 . . . 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 603 . 2  |-  ( (
ph  /\  ( X  e.  A  /\  -.  X  e.  W ) )  -> 
( F `  X
)  =  Z )
161, 15sylan2b 462 1  |-  ( (
ph  /\  X  e.  ( A  \  W ) )  ->  ( F `  X )  =  Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446
This theorem is referenced by:  cantnfp1lem1  7626  cantnfp1lem3  7628  cantnflem1d  7636  cantnflem1  7637  cnfcom2lem  7650  gsumval3  15506  gsumcllem  15508  gsumzaddlem  15518  gsumzsplit  15521  gsumzmhm  15525  gsumzinv  15532  gsumsub  15534  gsumpt  15537  gsum2d  15538  dprdfinv  15569  dprdfadd  15570  dmdprdsplitlem  15587  dpjidcl  15608  gsumdixp  15707  psrbaglesupp  16425  psrbagaddcl  16427  psrbaglefi  16429  mplsubglem  16490  mpllsslem  16491  mplsubrglem  16494  mplmonmul  16519  mplcoe1  16520  mplcoe2  16522  mplbas2  16523  evlslem4  16556  evlslem2  16560  deg1mul3le  20031  lcomfsup  26738  uvcresum  27210  frlmsslsp  27216  frlmup1  27218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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