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Theorem suppssr 5675
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 3175 . 2  |-  ( X  e.  ( A  \  W )  <->  ( X  e.  A  /\  -.  X  e.  W ) )
2 fvex 5555 . . . . . 6  |-  ( F `
 X )  e. 
_V
3 eldifsn 3762 . . . . . 6  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( ( F `  X )  e.  _V  /\  ( F `
 X )  =/= 
Z ) )
42, 3mpbiran 884 . . . . 5  |-  ( ( F `  X )  e.  ( _V  \  { Z } )  <->  ( F `  X )  =/=  Z
)
5 suppssr.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
6 ffn 5405 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
7 elpreima 5661 . . . . . . . 8  |-  ( F  Fn  A  ->  ( X  e.  ( `' F " ( _V  \  { Z } ) )  <-> 
( X  e.  A  /\  ( F `  X
)  e.  ( _V 
\  { Z }
) ) ) )
85, 6, 73syl 18 . . . . . . 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 3192 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  { Z }
) )  ->  X  e.  W ) )
118, 10sylbird 226 . . . . . 6  |-  ( ph  ->  ( ( X  e.  A  /\  ( F `
 X )  e.  ( _V  \  { Z } ) )  ->  X  e.  W )
)
1211expdimp 426 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
)  e.  ( _V 
\  { Z }
)  ->  X  e.  W ) )
134, 12syl5bir 209 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
)  =/=  Z  ->  X  e.  W )
)
1413necon1bd 2527 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( -.  X  e.  W  ->  ( F `  X
)  =  Z ) )
1514impr 602 . 2  |-  ( (
ph  /\  ( X  e.  A  /\  -.  X  e.  W ) )  -> 
( F `  X
)  =  Z )
161, 15sylan2b 461 1  |-  ( (
ph  /\  X  e.  ( A  \  W ) )  ->  ( F `  X )  =  Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271
This theorem is referenced by:  cantnfp1lem1  7396  cantnfp1lem3  7398  cantnflem1d  7406  cantnflem1  7407  cnfcom2lem  7420  gsumval3  15207  gsumcllem  15209  gsumzaddlem  15219  gsumzsplit  15222  gsumzmhm  15226  gsumzinv  15233  gsumsub  15235  gsumpt  15238  gsum2d  15239  dprdfinv  15270  dprdfadd  15271  dmdprdsplitlem  15288  dpjidcl  15309  gsumdixp  15408  psrbaglesupp  16130  psrbagaddcl  16132  psrbaglefi  16134  mplsubglem  16195  mpllsslem  16196  mplsubrglem  16199  mplmonmul  16224  mplcoe1  16225  mplcoe2  16227  mplbas2  16228  evlslem4  16261  evlslem2  16265  deg1mul3le  19518  lcomfsup  26871  uvcresum  27345  frlmsslsp  27351  frlmup1  27353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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