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Theorem suppss2f 25777
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 2433 . . 3  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
21mptpreima 5319 . 2  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
) )  =  {
k  e.  A  |  B  e.  ( _V  \  { Z } ) }
3 suppss2f.a . . . 4  |-  F/_ k A
4 nfcv 2569 . . . 4  |-  F/_ l A
5 nfv 1672 . . . 4  |-  F/ l  B  e.  ( _V 
\  { Z }
)
6 nfcsb1v 3292 . . . . 5  |-  F/_ k [_ l  /  k ]_ B
76nfel1 2579 . . . 4  |-  F/ k
[_ l  /  k ]_ B  e.  ( _V  \  { Z }
)
8 csbeq1a 3285 . . . . 5  |-  ( k  =  l  ->  B  =  [_ l  /  k ]_ B )
98eleq1d 2499 . . . 4  |-  ( k  =  l  ->  ( B  e.  ( _V  \  { Z } )  <->  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) ) )
103, 4, 5, 7, 9cbvrab 2960 . . 3  |-  { k  e.  A  |  B  e.  ( _V  \  { Z } ) }  =  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }
11 eldifsni 3989 . . . . . 6  |-  ( [_ l  /  k ]_ B  e.  ( _V  \  { Z } )  ->  [_ l  /  k ]_ B  =/=  Z )
12 eldif 3326 . . . . . . . . 9  |-  ( l  e.  ( A  \  W )  <->  ( l  e.  A  /\  -.  l  e.  W ) )
13 suppss2f.n . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
1413sbt 2119 . . . . . . . . . 10  |-  [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W
) )  ->  B  =  Z )
15 sbim 2085 . . . . . . . . . . 11  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
) )
16 sban 2089 . . . . . . . . . . . . 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 2068 . . . . . . . . . . . . . 14  |-  ( [ l  /  k ]
ph 
<-> 
ph )
19 suppss2f.w . . . . . . . . . . . . . . . 16  |-  F/_ k W
203, 19nfdif 3465 . . . . . . . . . . . . . . 15  |-  F/_ k
( A  \  W
)
2120clelsb3f 25686 . . . . . . . . . . . . . 14  |-  ( [ l  /  k ] k  e.  ( A 
\  W )  <->  l  e.  ( A  \  W ) )
2218, 21anbi12i 690 . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . 13  |-  ( [ l  /  k ] B  =  Z  <->  [. l  / 
k ]. B  =  Z )
25 vex 2965 . . . . . . . . . . . . . 14  |-  l  e. 
_V
26 sbceq1g 3670 . . . . . . . . . . . . . 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 326 . . . . . . . . . . 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 473 . . . . . . . 8  |-  ( (
ph  /\  ( l  e.  A  /\  -.  l  e.  W ) )  ->  [_ l  /  k ]_ B  =  Z
)
3332expr 610 . . . . . . 7  |-  ( (
ph  /\  l  e.  A )  ->  ( -.  l  e.  W  ->  [_ l  /  k ]_ B  =  Z
) )
3433necon1ad 2668 . . . . . 6  |-  ( (
ph  /\  l  e.  A )  ->  ( [_ l  /  k ]_ B  =/=  Z  ->  l  e.  W ) )
3511, 34syl5 32 . . . . 5  |-  ( (
ph  /\  l  e.  A )  ->  ( [_ l  /  k ]_ B  e.  ( _V  \  { Z }
)  ->  l  e.  W ) )
3635ss2rabdv 3421 . . . 4  |-  ( ph  ->  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }  C_  { l  e.  A  |  l  e.  W } )
37 dfin5 3324 . . . . 5  |-  ( A  i^i  W )  =  { l  e.  A  |  l  e.  W }
38 inss2 3559 . . . . 5  |-  ( A  i^i  W )  C_  W
3937, 38eqsstr3i 3375 . . . 4  |-  { l  e.  A  |  l  e.  W }  C_  W
4036, 39syl6ss 3356 . . 3  |-  ( ph  ->  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }  C_  W
)
4110, 40syl5eqss 3388 . 2  |-  ( ph  ->  { k  e.  A  |  B  e.  ( _V  \  { Z }
) }  C_  W
)
422, 41syl5eqss 3388 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 369    = wceq 1362   F/wnf 1592   [wsb 1699    e. wcel 1755   F/_wnfc 2556    =/= wne 2596   {crab 2709   _Vcvv 2962   [.wsbc 3175   [_csb 3276    \ cdif 3313    i^i cin 3315    C_ wss 3316   {csn 3865    e. cmpt 4338   `'ccnv 4826   "cima 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-mpt 4340  df-xp 4833  df-rel 4834  df-cnv 4835  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840
This theorem is referenced by:  esumss  26374
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