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Theorem suppss2f 27920
Description: Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.)
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 )
Assertion
Ref Expression
suppss2f  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { Z } ) ) 
C_  W )
Distinct variable group:    k, Z
Allowed substitution hints:    ph( k)    A( k)    B( k)    W( k)

Proof of Theorem suppss2f
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
21mptpreima 5316 . 2  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
) )  =  {
k  e.  A  |  B  e.  ( _V  \  { Z } ) }
3 suppss2f.a . . . 4  |-  F/_ k A
4 nfcv 2564 . . . 4  |-  F/_ l A
5 nfv 1728 . . . 4  |-  F/ l  B  e.  ( _V 
\  { Z }
)
6 nfcsb1v 3389 . . . . 5  |-  F/_ k [_ l  /  k ]_ B
76nfel1 2580 . . . 4  |-  F/ k
[_ l  /  k ]_ B  e.  ( _V  \  { Z }
)
8 csbeq1a 3382 . . . . 5  |-  ( k  =  l  ->  B  =  [_ l  /  k ]_ B )
98eleq1d 2471 . . . 4  |-  ( k  =  l  ->  ( B  e.  ( _V  \  { Z } )  <->  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) ) )
103, 4, 5, 7, 9cbvrab 3057 . . 3  |-  { k  e.  A  |  B  e.  ( _V  \  { Z } ) }  =  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }
11 eldifsni 4098 . . . . . 6  |-  ( [_ l  /  k ]_ B  e.  ( _V  \  { Z } )  ->  [_ l  /  k ]_ B  =/=  Z )
12 eldif 3424 . . . . . . . . 9  |-  ( l  e.  ( A  \  W )  <->  ( l  e.  A  /\  -.  l  e.  W ) )
13 suppss2f.n . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
1413sbt 2186 . . . . . . . . . 10  |-  [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W
) )  ->  B  =  Z )
15 sbim 2160 . . . . . . . . . . 11  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
) )
16 sban 2164 . . . . . . . . . . . . 13  |-  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W
) )  <->  ( [
l  /  k ]
ph  /\  [ l  /  k ] k  e.  ( A  \  W ) ) )
17 suppss2f.p . . . . . . . . . . . . . . 15  |-  F/ k
ph
1817sbf 2145 . . . . . . . . . . . . . 14  |-  ( [ l  /  k ]
ph 
<-> 
ph )
19 suppss2f.w . . . . . . . . . . . . . . . 16  |-  F/_ k W
203, 19nfdif 3564 . . . . . . . . . . . . . . 15  |-  F/_ k
( A  \  W
)
2120clelsb3f 27794 . . . . . . . . . . . . . 14  |-  ( [ l  /  k ] k  e.  ( A 
\  W )  <->  l  e.  ( A  \  W ) )
2218, 21anbi12i 695 . . . . . . . . . . . . 13  |-  ( ( [ l  /  k ] ph  /\  [ l  /  k ] k  e.  ( A  \  W ) )  <->  ( ph  /\  l  e.  ( A 
\  W ) ) )
2316, 22bitri 249 . . . . . . . . . . . 12  |-  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W
) )  <->  ( ph  /\  l  e.  ( A 
\  W ) ) )
24 sbsbc 3281 . . . . . . . . . . . . 13  |-  ( [ l  /  k ] B  =  Z  <->  [. l  / 
k ]. B  =  Z )
25 vex 3062 . . . . . . . . . . . . . 14  |-  l  e. 
_V
26 sbceq1g 3782 . . . . . . . . . . . . . 14  |-  ( l  e.  _V  ->  ( [. l  /  k ]. B  =  Z  <->  [_ l  /  k ]_ B  =  Z )
)
2725, 26ax-mp 5 . . . . . . . . . . . . 13  |-  ( [. l  /  k ]. B  =  Z  <->  [_ l  /  k ]_ B  =  Z
)
2824, 27bitri 249 . . . . . . . . . . . 12  |-  ( [ l  /  k ] B  =  Z  <->  [_ l  / 
k ]_ B  =  Z )
2923, 28imbi12i 324 . . . . . . . . . . 11  |-  ( ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
)  <->  ( ( ph  /\  l  e.  ( A 
\  W ) )  ->  [_ l  /  k ]_ B  =  Z
) )
3015, 29bitri 249 . . . . . . . . . 10  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( ( ph  /\  l  e.  ( A  \  W
) )  ->  [_ l  /  k ]_ B  =  Z ) )
3114, 30mpbi 208 . . . . . . . . 9  |-  ( (
ph  /\  l  e.  ( A  \  W ) )  ->  [_ l  / 
k ]_ B  =  Z )
3212, 31sylan2br 474 . . . . . . . 8  |-  ( (
ph  /\  ( l  e.  A  /\  -.  l  e.  W ) )  ->  [_ l  /  k ]_ B  =  Z
)
3332expr 613 . . . . . . 7  |-  ( (
ph  /\  l  e.  A )  ->  ( -.  l  e.  W  ->  [_ l  /  k ]_ B  =  Z
) )
3433necon1ad 2619 . . . . . 6  |-  ( (
ph  /\  l  e.  A )  ->  ( [_ l  /  k ]_ B  =/=  Z  ->  l  e.  W ) )
3511, 34syl5 30 . . . . 5  |-  ( (
ph  /\  l  e.  A )  ->  ( [_ l  /  k ]_ B  e.  ( _V  \  { Z }
)  ->  l  e.  W ) )
3635ss2rabdv 3520 . . . 4  |-  ( ph  ->  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }  C_  { l  e.  A  |  l  e.  W } )
37 dfin5 3422 . . . . 5  |-  ( A  i^i  W )  =  { l  e.  A  |  l  e.  W }
38 inss2 3660 . . . . 5  |-  ( A  i^i  W )  C_  W
3937, 38eqsstr3i 3473 . . . 4  |-  { l  e.  A  |  l  e.  W }  C_  W
4036, 39syl6ss 3454 . . 3  |-  ( ph  ->  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }  C_  W
)
4110, 40syl5eqss 3486 . 2  |-  ( ph  ->  { k  e.  A  |  B  e.  ( _V  \  { Z }
) }  C_  W
)
422, 41syl5eqss 3486 1  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { Z } ) ) 
C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   F/wnf 1637   [wsb 1763    e. wcel 1842   F/_wnfc 2550    =/= wne 2598   {crab 2758   _Vcvv 3059   [.wsbc 3277   [_csb 3373    \ cdif 3411    i^i cin 3413    C_ wss 3414   {csn 3972    |-> cmpt 4453   `'ccnv 4822   "cima 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-mpt 4455  df-xp 4829  df-rel 4830  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836
This theorem is referenced by:  esumss  28519
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