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Theorem suppss2OLD 6314
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 6722 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 2441 . . 3  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
21mptpreima 5328 . 2  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
) )  =  {
k  e.  A  |  B  e.  ( _V  \  { Z } ) }
3 eldifsni 3998 . . . . 5  |-  ( B  e.  ( _V  \  { Z } )  ->  B  =/=  Z )
4 eldif 3335 . . . . . . . 8  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
5 suppss2OLD.n . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
64, 5sylan2br 473 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  ->  B  =  Z )
76expr 612 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  B  =  Z ) )
87necon1ad 2676 . . . . 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 1179 . . 3  |-  ( (
ph  /\  k  e.  A  /\  B  e.  ( _V  \  { Z } ) )  -> 
k  e.  W )
1110rabssdv 3429 . 2  |-  ( ph  ->  { k  e.  A  |  B  e.  ( _V  \  { Z }
) }  C_  W
)
122, 11syl5eqss 3397 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 1364    e. wcel 1761    =/= wne 2604   {crab 2717   _Vcvv 2970    \ cdif 3322    C_ wss 3325   {csn 3874    e. cmpt 4347   `'ccnv 4835   "cima 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-mpt 4349  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849
This theorem is referenced by:  cantnflem1dOLD  7915  cantnflem1OLD  7916  gsumzsplitOLD  16412  gsum2dOLD  16454  dprdfidOLD  16504  dprdfinvOLD  16506  dprdfaddOLD  16507  dmdprdsplitlemOLD  16525  dpjidclOLD  16554  psrbagaddclOLD  17417  psrbasOLD  17427  psrlidmOLD  17453  psrridmOLD  17455  mvridlemOLD  17470  mplcoe3OLD  17524  mplcoe2OLD  17526  mplbas2OLD  17528  evlslem4OLD  17566
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