MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppss2OLD Structured version   Unicode version

Theorem suppss2OLD 6503
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 6926 as of 28-May-2019. (New usage is discouraged.) (Proof modification 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 2454 . . 3  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
21mptpreima 5483 . 2  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
) )  =  {
k  e.  A  |  B  e.  ( _V  \  { Z } ) }
3 eldifsni 4142 . . . . 5  |-  ( B  e.  ( _V  \  { Z } )  ->  B  =/=  Z )
4 eldif 3471 . . . . . . . 8  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
5 suppss2OLD.n . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
64, 5sylan2br 474 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  A  /\  -.  k  e.  W ) )  ->  B  =  Z )
76expr 613 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  B  =  Z ) )
87necon1ad 2670 . . . . 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 1191 . . 3  |-  ( (
ph  /\  k  e.  A  /\  B  e.  ( _V  \  { Z } ) )  -> 
k  e.  W )
1110rabssdv 3566 . 2  |-  ( ph  ->  { k  e.  A  |  B  e.  ( _V  \  { Z }
) }  C_  W
)
122, 11syl5eqss 3533 1  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { Z } ) ) 
C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016    |-> cmpt 4497   `'ccnv 4987   "cima 4991
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-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-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  cantnflem1dOLD  8121  cantnflem1OLD  8122  gsumzsplitOLD  17147  gsum2dOLD  17199  dprdfidOLD  17262  dprdfinvOLD  17264  dprdfaddOLD  17265  dmdprdsplitlemOLD  17283  dpjidclOLD  17312  psrbagaddclOLD  18219  psrbasOLD  18229  psrlidmOLD  18255  psrridmOLD  18257  mvridlemOLD  18273  mplcoe3OLD  18327  mplcoe2OLD  18331  mplbas2OLD  18333  evlslem4OLD  18371
  Copyright terms: Public domain W3C validator