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Theorem suppssOLD 5996
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6922 as of 28-May-2019. (New usage is discouraged.) (Proof modification 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 5713 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
3 elpreima 5983 . . . 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 5858 . . . . . 6  |-  ( F `
 k )  e. 
_V
6 eldifsn 4141 . . . . . 6  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( ( F `  k )  e.  _V  /\  ( F `
 k )  =/= 
Z ) )
75, 6mpbiran 916 . . . . 5  |-  ( ( F `  k )  e.  ( _V  \  { Z } )  <->  ( F `  k )  =/=  Z
)
8 eldif 3471 . . . . . . . 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 474 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  -> 
( F `  k
)  =  Z )
1110expr 613 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  ( F `  k
)  =  Z ) )
1211necon1ad 2670 . . . . 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 601 . . 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 3495 1  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016   `'ccnv 4987   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  cantnfp1lem1OLD  8114  cantnfp1lem3OLD  8116  gsumzaddlemOLD  17135  gsumzmhmOLD  17156  gsumzinvOLD  17168  lcomfsupOLD  17744  psrbaglesuppOLD  18213  psrlidmOLD  18252  psrridmOLD  18254  mplsubglemOLD  18290  mpllsslemOLD  18291  mplsubrglemOLD  18296
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