Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  suppss2f Structured version   Visualization version   Unicode version

Theorem suppss2f 28314
Description: Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)
Hypotheses
Ref Expression
suppss2f.p  |-  F/ k
ph
suppss2f.a  |-  F/_ k A
suppss2f.w  |-  F/_ k W
suppss2f.n  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
suppss2f.v  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
suppss2f  |-  ( ph  ->  ( ( k  e.  A  |->  B ) supp  Z
)  C_  W )
Distinct variable group:    k, Z
Allowed substitution hints:    ph( k)    A( k)    B( k)    V( k)    W( k)

Proof of Theorem suppss2f
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 suppss2f.a . . . 4  |-  F/_ k A
2 nfcv 2612 . . . 4  |-  F/_ l A
3 nfcv 2612 . . . 4  |-  F/_ l B
4 nfcsb1v 3365 . . . 4  |-  F/_ k [_ l  /  k ]_ B
5 csbeq1a 3358 . . . 4  |-  ( k  =  l  ->  B  =  [_ l  /  k ]_ B )
61, 2, 3, 4, 5cbvmptf 4486 . . 3  |-  ( k  e.  A  |->  B )  =  ( l  e.  A  |->  [_ l  /  k ]_ B )
76oveq1i 6318 . 2  |-  ( ( k  e.  A  |->  B ) supp  Z )  =  ( ( l  e.  A  |->  [_ l  /  k ]_ B ) supp  Z )
8 suppss2f.n . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
98sbt 2268 . . . 4  |-  [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W
) )  ->  B  =  Z )
10 sbim 2244 . . . . 5  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
) )
11 sban 2248 . . . . . . 7  |-  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W
) )  <->  ( [
l  /  k ]
ph  /\  [ l  /  k ] k  e.  ( A  \  W ) ) )
12 suppss2f.p . . . . . . . . 9  |-  F/ k
ph
1312sbf 2229 . . . . . . . 8  |-  ( [ l  /  k ]
ph 
<-> 
ph )
14 suppss2f.w . . . . . . . . . 10  |-  F/_ k W
151, 14nfdif 3543 . . . . . . . . 9  |-  F/_ k
( A  \  W
)
1615clelsb3f 28195 . . . . . . . 8  |-  ( [ l  /  k ] k  e.  ( A 
\  W )  <->  l  e.  ( A  \  W ) )
1713, 16anbi12i 711 . . . . . . 7  |-  ( ( [ l  /  k ] ph  /\  [ l  /  k ] k  e.  ( A  \  W ) )  <->  ( ph  /\  l  e.  ( A 
\  W ) ) )
1811, 17bitri 257 . . . . . 6  |-  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W
) )  <->  ( ph  /\  l  e.  ( A 
\  W ) ) )
19 sbsbc 3259 . . . . . . 7  |-  ( [ l  /  k ] B  =  Z  <->  [. l  / 
k ]. B  =  Z )
20 vex 3034 . . . . . . . 8  |-  l  e. 
_V
21 sbceq1g 3781 . . . . . . . 8  |-  ( l  e.  _V  ->  ( [. l  /  k ]. B  =  Z  <->  [_ l  /  k ]_ B  =  Z )
)
2220, 21ax-mp 5 . . . . . . 7  |-  ( [. l  /  k ]. B  =  Z  <->  [_ l  /  k ]_ B  =  Z
)
2319, 22bitri 257 . . . . . 6  |-  ( [ l  /  k ] B  =  Z  <->  [_ l  / 
k ]_ B  =  Z )
2418, 23imbi12i 333 . . . . 5  |-  ( ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
)  <->  ( ( ph  /\  l  e.  ( A 
\  W ) )  ->  [_ l  /  k ]_ B  =  Z
) )
2510, 24bitri 257 . . . 4  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( ( ph  /\  l  e.  ( A  \  W
) )  ->  [_ l  /  k ]_ B  =  Z ) )
269, 25mpbi 213 . . 3  |-  ( (
ph  /\  l  e.  ( A  \  W ) )  ->  [_ l  / 
k ]_ B  =  Z )
27 suppss2f.v . . 3  |-  ( ph  ->  A  e.  V )
2826, 27suppss2 6968 . 2  |-  ( ph  ->  ( ( l  e.  A  |->  [_ l  /  k ]_ B ) supp  Z ) 
C_  W )
297, 28syl5eqss 3462 1  |-  ( ph  ->  ( ( k  e.  A  |->  B ) supp  Z
)  C_  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   F/wnf 1675   [wsb 1805    e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   [.wsbc 3255   [_csb 3349    \ cdif 3387    C_ wss 3390    |-> cmpt 4454  (class class class)co 6308   supp csupp 6933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-supp 6934
This theorem is referenced by:  esumss  28967
  Copyright terms: Public domain W3C validator