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Theorem suppss2fOLD 28239
Description: Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) Obsolete version of suppss2f 28240 as of 1-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
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
suppss2fOLD  |-  ( 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 suppss2fOLD
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 eqid 2422 . . 3  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
21mptpreima 5347 . 2  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
) )  =  {
k  e.  A  |  B  e.  ( _V  \  { Z } ) }
3 suppss2f.a . . . 4  |-  F/_ k A
4 nfcv 2580 . . . 4  |-  F/_ l A
5 nfv 1755 . . . 4  |-  F/ l  B  e.  ( _V 
\  { Z }
)
6 nfcsb1v 3411 . . . . 5  |-  F/_ k [_ l  /  k ]_ B
76nfel1 2596 . . . 4  |-  F/ k
[_ l  /  k ]_ B  e.  ( _V  \  { Z }
)
8 csbeq1a 3404 . . . . 5  |-  ( k  =  l  ->  B  =  [_ l  /  k ]_ B )
98eleq1d 2491 . . . 4  |-  ( k  =  l  ->  ( B  e.  ( _V  \  { Z } )  <->  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) ) )
103, 4, 5, 7, 9cbvrab 3078 . . 3  |-  { k  e.  A  |  B  e.  ( _V  \  { Z } ) }  =  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }
11 eldifsni 4126 . . . . . 6  |-  ( [_ l  /  k ]_ B  e.  ( _V  \  { Z } )  ->  [_ l  /  k ]_ B  =/=  Z )
12 eldif 3446 . . . . . . . . 9  |-  ( l  e.  ( A  \  W )  <->  ( l  e.  A  /\  -.  l  e.  W ) )
13 suppss2f.n . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )
1413sbt 2217 . . . . . . . . . 10  |-  [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W
) )  ->  B  =  Z )
15 sbim 2193 . . . . . . . . . . 11  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
) )
16 sban 2197 . . . . . . . . . . . . 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 2178 . . . . . . . . . . . . . 14  |-  ( [ l  /  k ]
ph 
<-> 
ph )
19 suppss2f.w . . . . . . . . . . . . . . . 16  |-  F/_ k W
203, 19nfdif 3586 . . . . . . . . . . . . . . 15  |-  F/_ k
( A  \  W
)
2120clelsb3f 28114 . . . . . . . . . . . . . 14  |-  ( [ l  /  k ] k  e.  ( A 
\  W )  <->  l  e.  ( A  \  W ) )
2218, 21anbi12i 701 . . . . . . . . . . . . 13  |-  ( ( [ l  /  k ] ph  /\  [ l  /  k ] k  e.  ( A  \  W ) )  <->  ( ph  /\  l  e.  ( A 
\  W ) ) )
2316, 22bitri 252 . . . . . . . . . . . 12  |-  ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W
) )  <->  ( ph  /\  l  e.  ( A 
\  W ) ) )
24 sbsbc 3303 . . . . . . . . . . . . 13  |-  ( [ l  /  k ] B  =  Z  <->  [. l  / 
k ]. B  =  Z )
25 vex 3083 . . . . . . . . . . . . . 14  |-  l  e. 
_V
26 sbceq1g 3807 . . . . . . . . . . . . . 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 252 . . . . . . . . . . . 12  |-  ( [ l  /  k ] B  =  Z  <->  [_ l  / 
k ]_ B  =  Z )
2923, 28imbi12i 327 . . . . . . . . . . 11  |-  ( ( [ l  /  k ] ( ph  /\  k  e.  ( A  \  W ) )  ->  [ l  /  k ] B  =  Z
)  <->  ( ( ph  /\  l  e.  ( A 
\  W ) )  ->  [_ l  /  k ]_ B  =  Z
) )
3015, 29bitri 252 . . . . . . . . . 10  |-  ( [ l  /  k ] ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )  <->  ( ( ph  /\  l  e.  ( A  \  W
) )  ->  [_ l  /  k ]_ B  =  Z ) )
3114, 30mpbi 211 . . . . . . . . 9  |-  ( (
ph  /\  l  e.  ( A  \  W ) )  ->  [_ l  / 
k ]_ B  =  Z )
3212, 31sylan2br 478 . . . . . . . 8  |-  ( (
ph  /\  ( l  e.  A  /\  -.  l  e.  W ) )  ->  [_ l  /  k ]_ B  =  Z
)
3332expr 618 . . . . . . 7  |-  ( (
ph  /\  l  e.  A )  ->  ( -.  l  e.  W  ->  [_ l  /  k ]_ B  =  Z
) )
3433necon1ad 2636 . . . . . 6  |-  ( (
ph  /\  l  e.  A )  ->  ( [_ l  /  k ]_ B  =/=  Z  ->  l  e.  W ) )
3511, 34syl5 33 . . . . 5  |-  ( (
ph  /\  l  e.  A )  ->  ( [_ l  /  k ]_ B  e.  ( _V  \  { Z }
)  ->  l  e.  W ) )
3635ss2rabdv 3542 . . . 4  |-  ( ph  ->  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }  C_  { l  e.  A  |  l  e.  W } )
37 dfin5 3444 . . . . 5  |-  ( A  i^i  W )  =  { l  e.  A  |  l  e.  W }
38 inss2 3683 . . . . 5  |-  ( A  i^i  W )  C_  W
3937, 38eqsstr3i 3495 . . . 4  |-  { l  e.  A  |  l  e.  W }  C_  W
4036, 39syl6ss 3476 . . 3  |-  ( ph  ->  { l  e.  A  |  [_ l  /  k ]_ B  e.  ( _V  \  { Z }
) }  C_  W
)
4110, 40syl5eqss 3508 . 2  |-  ( ph  ->  { k  e.  A  |  B  e.  ( _V  \  { Z }
) }  C_  W
)
422, 41syl5eqss 3508 1  |-  ( ph  ->  ( `' ( k  e.  A  |->  B )
" ( _V  \  { Z } ) ) 
C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   F/wnf 1661   [wsb 1790    e. wcel 1872   F/_wnfc 2566    =/= wne 2614   {crab 2775   _Vcvv 3080   [.wsbc 3299   [_csb 3395    \ cdif 3433    i^i cin 3435    C_ wss 3436   {csn 3998    |-> cmpt 4482   `'ccnv 4852   "cima 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-mpt 4484  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866
This theorem is referenced by: (None)
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