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Theorem suppss 6930
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
Hypotheses
Ref Expression
suppss.f  |-  ( ph  ->  F : A --> B )
suppss.n  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
Assertion
Ref Expression
suppss  |-  ( ph  ->  ( F supp  Z ) 
C_  W )
Distinct variable groups:    k, F    ph, k    k, W    k, Z
Allowed substitution hints:    A( k)    B( k)

Proof of Theorem suppss
StepHypRef Expression
1 suppss.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
2 ffn 5730 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43adantl 466 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  F  Fn  A )
5 fdm 5734 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
6 dmexg 6715 . . . . . . . . . 10  |-  ( F  e.  _V  ->  dom  F  e.  _V )
76adantr 465 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  dom  F  e.  _V )
8 eleq1 2539 . . . . . . . . . 10  |-  ( A  =  dom  F  -> 
( A  e.  _V  <->  dom 
F  e.  _V )
)
98eqcoms 2479 . . . . . . . . 9  |-  ( dom 
F  =  A  -> 
( A  e.  _V  <->  dom 
F  e.  _V )
)
107, 9syl5ibr 221 . . . . . . . 8  |-  ( dom 
F  =  A  -> 
( ( F  e. 
_V  /\  Z  e.  _V )  ->  A  e. 
_V ) )
111, 5, 103syl 20 . . . . . . 7  |-  ( ph  ->  ( ( F  e. 
_V  /\  Z  e.  _V )  ->  A  e. 
_V ) )
1211impcom 430 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  A  e.  _V )
13 simplr 754 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  Z  e.  _V )
14 elsuppfn 6909 . . . . . 6  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  Z  e.  _V )  ->  (
k  e.  ( F supp 
Z )  <->  ( k  e.  A  /\  ( F `  k )  =/=  Z ) ) )
154, 12, 13, 14syl3anc 1228 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
k  e.  ( F supp 
Z )  <->  ( k  e.  A  /\  ( F `  k )  =/=  Z ) ) )
16 eldif 3486 . . . . . . . . 9  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
17 suppss.n . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
1817adantll 713 . . . . . . . . 9  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
1916, 18sylan2br 476 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  ( k  e.  A  /\  -.  k  e.  W
) )  ->  ( F `  k )  =  Z )
2019expr 615 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  A
)  ->  ( -.  k  e.  W  ->  ( F `  k )  =  Z ) )
2120necon1ad 2683 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  A
)  ->  ( ( F `  k )  =/=  Z  ->  k  e.  W ) )
2221expimpd 603 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
( k  e.  A  /\  ( F `  k
)  =/=  Z )  ->  k  e.  W
) )
2315, 22sylbid 215 . . . 4  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
k  e.  ( F supp 
Z )  ->  k  e.  W ) )
2423ssrdv 3510 . . 3  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F supp  Z )  C_  W
)
2524ex 434 . 2  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  ( ph  ->  ( F supp  Z )  C_  W
) )
26 supp0prc 6904 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
27 0ss 3814 . . . 4  |-  (/)  C_  W
2826, 27syl6eqss 3554 . . 3  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z ) 
C_  W )
2928a1d 25 . 2  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( ph  ->  ( F supp  Z )  C_  W
) )
3025, 29pm2.61i 164 1  |-  ( ph  ->  ( F supp  Z ) 
C_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   dom cdm 4999    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   supp csupp 6901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-supp 6902
This theorem is referenced by:  fsuppco2  7861  fsuppcor  7862  cantnfp1lem1  8096  cantnfp1lem3  8098  gsumzaddlem  16734  gsumzmhm  16757  gsum2d2lem  16801  lcomfsupp  17345  psrbaglesupp  17804  mplsubglem  17880  mpllsslem  17881  mplsubrglem  17887  mvrcl  17898  evlslem3  17970  frlmssuvc1  18608  frlmsslsp  18612  frlmup2  18616  deg1mul3le  22268  jensen  23062  resf1o  27241
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