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Theorem suppssOLD 5932
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6816 as of 28-May-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
suppssOLD.f  |-  ( ph  ->  F : A --> B )
suppssOLD.n  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
Assertion
Ref Expression
suppssOLD  |-  ( 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 suppssOLD
StepHypRef Expression
1 suppssOLD.f . . . 4  |-  ( ph  ->  F : A --> B )
2 ffn 5654 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
3 elpreima 5919 . . . 4  |-  ( F  Fn  A  ->  (
k  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( k  e.  A  /\  ( F `  k )  e.  ( _V  \  { Z } ) ) ) )
41, 2, 33syl 20 . . 3  |-  ( ph  ->  ( k  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( k  e.  A  /\  ( F `  k )  e.  ( _V  \  { Z } ) ) ) )
5 fvex 5796 . . . . . 6  |-  ( F `
 k )  e. 
_V
6 eldifsn 4095 . . . . . 6  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( ( F `  k )  e.  _V  /\  ( F `
 k )  =/= 
Z ) )
75, 6mpbiran 909 . . . . 5  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( F `  k )  =/=  Z
)
8 eldif 3433 . . . . . . . 8  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
9 suppssOLD.n . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
108, 9sylan2br 476 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  -> 
( F `  k
)  =  Z )
1110expr 615 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  ( F `  k
)  =  Z ) )
1211necon1ad 2662 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  =/=  Z  -> 
k  e.  W ) )
137, 12syl5bi 217 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  e.  ( _V 
\  { Z }
)  ->  k  e.  W ) )
1413expimpd 603 . . 3  |-  ( ph  ->  ( ( k  e.  A  /\  ( F `
 k )  e.  ( _V  \  { Z } ) )  -> 
k  e.  W ) )
154, 14sylbid 215 . 2  |-  ( ph  ->  ( k  e.  ( `' F " ( _V 
\  { Z }
) )  ->  k  e.  W ) )
1615ssrdv 3457 1  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642   _Vcvv 3065    \ cdif 3420    C_ wss 3423   {csn 3972   `'ccnv 4934   "cima 4938    Fn wfn 5508   -->wf 5509   ` cfv 5513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521
This theorem is referenced by:  cantnfp1lem1OLD  8010  cantnfp1lem3OLD  8012  gsumzaddlemOLD  16511  gsumzmhmOLD  16533  gsumzinvOLD  16545  gsumsubOLD  16550  lcomfsupOLD  17087  psrbaglesuppOLD  17539  psrlidmOLD  17578  psrridmOLD  17580  mplsubglemOLD  17616  mpllsslemOLD  17617  mplsubrglemOLD  17622  frlmssuvc1OLD  18327  frlmsslspOLD  18330
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