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Theorem suppssr 5659
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 3162 . 2  |-  ( X  e.  ( A  \  W )  <->  ( X  e.  A  /\  -.  X  e.  W ) )
2 fvex 5539 . . . . . 6  |-  ( F `
 X )  e. 
_V
3 eldifsn 3749 . . . . . 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 5389 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
7 elpreima 5645 . . . . . . . 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 3179 . . . . . . 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 2514 . . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  cantnfp1lem1  7380  cantnfp1lem3  7382  cantnflem1d  7390  cantnflem1  7391  cnfcom2lem  7404  gsumval3  15191  gsumcllem  15193  gsumzaddlem  15203  gsumzsplit  15206  gsumzmhm  15210  gsumzinv  15217  gsumsub  15219  gsumpt  15222  gsum2d  15223  dprdfinv  15254  dprdfadd  15255  dmdprdsplitlem  15272  dpjidcl  15293  gsumdixp  15392  psrbaglesupp  16114  psrbagaddcl  16116  psrbaglefi  16118  mplsubglem  16179  mpllsslem  16180  mplsubrglem  16183  mplmonmul  16208  mplcoe1  16209  mplcoe2  16211  mplbas2  16212  evlslem4  16245  evlslem2  16249  deg1mul3le  19502  lcomfsup  26768  uvcresum  27242  frlmsslsp  27248  frlmup1  27250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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