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Theorem suppss 5795
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 5524 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
3 elpreima 5782 . . . 4  |-  ( F  Fn  A  ->  (
k  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( k  e.  A  /\  ( F `  k )  e.  ( _V  \  { Z } ) ) ) )
41, 2, 33syl 19 . . 3  |-  ( ph  ->  ( k  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( k  e.  A  /\  ( F `  k )  e.  ( _V  \  { Z } ) ) ) )
5 fvex 5675 . . . . . 6  |-  ( F `
 k )  e. 
_V
6 eldifsn 3863 . . . . . 6  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( ( F `  k )  e.  _V  /\  ( F `
 k )  =/= 
Z ) )
75, 6mpbiran 885 . . . . 5  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( F `  k )  =/=  Z
)
8 eldif 3266 . . . . . . . 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 463 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  -> 
( F `  k
)  =  Z )
1110expr 599 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  ( F `  k
)  =  Z ) )
1211necon1ad 2610 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  =/=  Z  -> 
k  e.  W ) )
137, 12syl5bi 209 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  e.  ( _V 
\  { Z }
)  ->  k  e.  W ) )
1413expimpd 587 . . 3  |-  ( ph  ->  ( ( k  e.  A  /\  ( F `
 k )  e.  ( _V  \  { Z } ) )  -> 
k  e.  W ) )
154, 14sylbid 207 . 2  |-  ( ph  ->  ( k  e.  ( `' F " ( _V 
\  { Z }
) )  ->  k  e.  W ) )
1615ssrdv 3290 1  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   _Vcvv 2892    \ cdif 3253    C_ wss 3256   {csn 3750   `'ccnv 4810   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387
This theorem is referenced by:  cantnfp1lem1  7560  cantnfp1lem3  7562  gsumzaddlem  15446  gsumzmhm  15453  gsumzinv  15460  gsumsub  15462  gsum2d2lem  15467  dprdsubg  15502  psrbaglesupp  16353  psrlidm  16387  psrridm  16388  mplsubglem  16418  mpllsslem  16419  mplsubrglem  16422  mvrcl  16432  evlslem3  19795  mdeg0  19853  deg1mul3le  19899  jensen  20687  lcomfsup  26431  frlmssuvc1  26908  frlmsslsp  26910  frlmup1  26912  frlmup2  26913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395
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