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Theorem suppss2 6924
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.) (Revised by AV, 28-May-2019.)
Hypotheses
Ref Expression
suppss2.n  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
suppss2.a  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
suppss2  |-  ( ph  ->  ( ( k  e.  A  |->  B ) supp  Z
)  C_  W )
Distinct variable groups:    A, k    ph, k    k, W    k, Z
Allowed substitution hints:    B( k)    V( k)

Proof of Theorem suppss2
StepHypRef Expression
1 eqid 2460 . . . . 5  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
2 suppss2.a . . . . . 6  |-  ( ph  ->  A  e.  V )
32adantl 466 . . . . 5  |-  ( ( Z  e.  _V  /\  ph )  ->  A  e.  V )
4 simpl 457 . . . . 5  |-  ( ( Z  e.  _V  /\  ph )  ->  Z  e.  _V )
51, 3, 4mptsuppdifd 6912 . . . 4  |-  ( ( Z  e.  _V  /\  ph )  ->  ( (
k  e.  A  |->  B ) supp  Z )  =  { k  e.  A  |  B  e.  ( _V  \  { Z }
) } )
6 eldifsni 4146 . . . . . . 7  |-  ( B  e.  ( _V  \  { Z } )  ->  B  =/=  Z )
7 eldif 3479 . . . . . . . . . 10  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
8 suppss2.n . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
98adantll 713 . . . . . . . . . 10  |-  ( ( ( Z  e.  _V  /\ 
ph )  /\  k  e.  ( A  \  W
) )  ->  B  =  Z )
107, 9sylan2br 476 . . . . . . . . 9  |-  ( ( ( Z  e.  _V  /\ 
ph )  /\  (
k  e.  A  /\  -.  k  e.  W
) )  ->  B  =  Z )
1110expr 615 . . . . . . . 8  |-  ( ( ( Z  e.  _V  /\ 
ph )  /\  k  e.  A )  ->  ( -.  k  e.  W  ->  B  =  Z ) )
1211necon1ad 2676 . . . . . . 7  |-  ( ( ( Z  e.  _V  /\ 
ph )  /\  k  e.  A )  ->  ( B  =/=  Z  ->  k  e.  W ) )
136, 12syl5 32 . . . . . 6  |-  ( ( ( Z  e.  _V  /\ 
ph )  /\  k  e.  A )  ->  ( B  e.  ( _V  \  { Z } )  ->  k  e.  W
) )
14133impia 1188 . . . . 5  |-  ( ( ( Z  e.  _V  /\ 
ph )  /\  k  e.  A  /\  B  e.  ( _V  \  { Z } ) )  -> 
k  e.  W )
1514rabssdv 3573 . . . 4  |-  ( ( Z  e.  _V  /\  ph )  ->  { k  e.  A  |  B  e.  ( _V  \  { Z } ) }  C_  W )
165, 15eqsstrd 3531 . . 3  |-  ( ( Z  e.  _V  /\  ph )  ->  ( (
k  e.  A  |->  B ) supp  Z )  C_  W )
1716ex 434 . 2  |-  ( Z  e.  _V  ->  ( ph  ->  ( ( k  e.  A  |->  B ) supp 
Z )  C_  W
) )
18 id 22 . . . . . 6  |-  ( -.  Z  e.  _V  ->  -.  Z  e.  _V )
1918intnand 909 . . . . 5  |-  ( -.  Z  e.  _V  ->  -.  ( ( k  e.  A  |->  B )  e. 
_V  /\  Z  e.  _V ) )
20 supp0prc 6894 . . . . 5  |-  ( -.  ( ( k  e.  A  |->  B )  e. 
_V  /\  Z  e.  _V )  ->  ( ( k  e.  A  |->  B ) supp  Z )  =  (/) )
2119, 20syl 16 . . . 4  |-  ( -.  Z  e.  _V  ->  ( ( k  e.  A  |->  B ) supp  Z )  =  (/) )
22 0ss 3807 . . . 4  |-  (/)  C_  W
2321, 22syl6eqss 3547 . . 3  |-  ( -.  Z  e.  _V  ->  ( ( k  e.  A  |->  B ) supp  Z ) 
C_  W )
2423a1d 25 . 2  |-  ( -.  Z  e.  _V  ->  (
ph  ->  ( ( k  e.  A  |->  B ) supp 
Z )  C_  W
) )
2517, 24pm2.61i 164 1  |-  ( ph  ->  ( ( k  e.  A  |->  B ) supp  Z
)  C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811   _Vcvv 3106    \ cdif 3466    C_ wss 3469   (/)c0 3778   {csn 4020    |-> cmpt 4498  (class class class)co 6275   supp csupp 6891
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-supp 6892
This theorem is referenced by:  suppsssn  6925  fsuppmptif  7848  sniffsupp  7858  cantnflem1d  8096  cantnflem1  8097  gsumzsplit  16728  gsummpt1n0  16776  gsum2dlem1  16781  gsum2dlem2  16782  gsum2d  16783  dprdfid  16840  dprdfinv  16842  dprdfadd  16843  dmdprdsplitlem  16867  dpjidcl  16890  psrbagaddcl  17784  psrlidm  17820  psrridm  17822  mplsubrg  17866  mplmon  17889  mplmonmul  17890  mplcoe1  17891  mplcoe5  17895  mplbas2  17898  evlslem4  17938  evlslem2  17944  evlslem3  17947  evlslem1  17948  coe1tmmul2  18081  coe1tmmul  18082  uvcff  18582  uvcresum  18584  tsmssplit  20382  coe1mul3  22228  plypf1  22337  tayl0  22484  suppss3  27208
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