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Theorem suppss2OLD 6420
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) Obsolete version of suppss2 6828 as of 28-May-2019. (New usage is discouraged.)
Hypothesis
Ref Expression
suppss2OLD.n  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
Assertion
Ref Expression
suppss2OLD  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { Z } ) ) 
C_  W )
Distinct variable groups:    A, k    ph, k    k, W    k, Z
Allowed substitution hint:    B( k)

Proof of Theorem suppss2OLD
StepHypRef Expression
1 eqid 2452 . . 3  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
21mptpreima 5434 . 2  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
) )  =  {
k  e.  A  |  B  e.  ( _V  \  { Z } ) }
3 eldifsni 4104 . . . . 5  |-  ( B  e.  ( _V  \  { Z } )  ->  B  =/=  Z )
4 eldif 3441 . . . . . . . 8  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
5 suppss2OLD.n . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
64, 5sylan2br 476 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  ->  B  =  Z )
76expr 615 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  B  =  Z ) )
87necon1ad 2665 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( B  =/=  Z  ->  k  e.  W ) )
93, 8syl5 32 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  ( B  e.  ( _V  \  { Z } )  ->  k  e.  W
) )
1093impia 1185 . . 3  |-  ( (
ph  /\  k  e.  A  /\  B  e.  ( _V  \  { Z } ) )  -> 
k  e.  W )
1110rabssdv 3535 . 2  |-  ( ph  ->  { k  e.  A  |  B  e.  ( _V  \  { Z }
) }  C_  W
)
122, 11syl5eqss 3503 1  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { Z } ) ) 
C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   {crab 2800   _Vcvv 3072    \ cdif 3428    C_ wss 3431   {csn 3980    |-> cmpt 4453   `'ccnv 4942   "cima 4946
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-mpt 4455  df-xp 4949  df-rel 4950  df-cnv 4951  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956
This theorem is referenced by:  cantnflem1dOLD  8025  cantnflem1OLD  8026  gsumzsplitOLD  16535  gsum2dOLD  16581  dprdfidOLD  16631  dprdfinvOLD  16633  dprdfaddOLD  16634  dmdprdsplitlemOLD  16652  dpjidclOLD  16681  psrbagaddclOLD  17557  psrbasOLD  17567  psrlidmOLD  17593  psrridmOLD  17595  mvridlemOLD  17611  mplcoe3OLD  17665  mplcoe2OLD  17669  mplbas2OLD  17671  evlslem4OLD  17709
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