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Theorem suppss 5658
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 5389 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
3 elpreima 5645 . . . 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 5539 . . . . . 6  |-  ( F `
 k )  e. 
_V
6 eldifsn 3749 . . . . . 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 3162 . . . . . . . 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 2513 . . . . 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 3185 1  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
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  gsumzaddlem  15203  gsumzmhm  15210  gsumzinv  15217  gsumsub  15219  gsum2d2lem  15224  dprdsubg  15259  psrbaglesupp  16114  psrlidm  16148  psrridm  16149  mplsubglem  16179  mpllsslem  16180  mplsubrglem  16183  mvrcl  16193  evlslem3  19398  mdeg0  19456  deg1mul3le  19502  jensen  20283  lcomfsup  26768  frlmssuvc1  27246  frlmsslsp  27248  frlmup1  27250  frlmup2  27251
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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