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Theorem suppss 5674
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
suppss.f  |-  ( ph  ->  F : A --> B )
suppss.n  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
Assertion
Ref Expression
suppss  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
Distinct variable groups:    k, F    ph, k    k, W    k, Z
Allowed substitution hints:    A( k)    B( k)

Proof of Theorem suppss
StepHypRef Expression
1 suppss.f . . . 4  |-  ( ph  ->  F : A --> B )
2 ffn 5405 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
3 elpreima 5661 . . . 4  |-  ( F  Fn  A  ->  (
k  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( k  e.  A  /\  ( F `  k )  e.  ( _V  \  { Z } ) ) ) )
41, 2, 33syl 18 . . 3  |-  ( ph  ->  ( k  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( k  e.  A  /\  ( F `  k )  e.  ( _V  \  { Z } ) ) ) )
5 fvex 5555 . . . . . 6  |-  ( F `
 k )  e. 
_V
6 eldifsn 3762 . . . . . 6  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( ( F `  k )  e.  _V  /\  ( F `
 k )  =/= 
Z ) )
75, 6mpbiran 884 . . . . 5  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( F `  k )  =/=  Z
)
8 eldif 3175 . . . . . . . 8  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
9 suppss.n . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
108, 9sylan2br 462 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  -> 
( F `  k
)  =  Z )
1110expr 598 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  ( F `  k
)  =  Z ) )
1211necon1ad 2526 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  =/=  Z  -> 
k  e.  W ) )
137, 12syl5bi 208 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  e.  ( _V 
\  { Z }
)  ->  k  e.  W ) )
1413expimpd 586 . . 3  |-  ( ph  ->  ( ( k  e.  A  /\  ( F `
 k )  e.  ( _V  \  { Z } ) )  -> 
k  e.  W ) )
154, 14sylbid 206 . 2  |-  ( ph  ->  ( k  e.  ( `' F " ( _V 
\  { Z }
) )  ->  k  e.  W ) )
1615ssrdv 3198 1  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
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  gsumzaddlem  15219  gsumzmhm  15226  gsumzinv  15233  gsumsub  15235  gsum2d2lem  15240  dprdsubg  15275  psrbaglesupp  16130  psrlidm  16164  psrridm  16165  mplsubglem  16195  mpllsslem  16196  mplsubrglem  16199  mvrcl  16209  evlslem3  19414  mdeg0  19472  deg1mul3le  19518  jensen  20299  lcomfsup  26871  frlmssuvc1  27349  frlmsslsp  27351  frlmup1  27353  frlmup2  27354
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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