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Theorem suppss 6731
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 5571 . . . . . . . 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 5575 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
6 dmexg 6521 . . . . . . . . . 10  |-  ( F  e.  _V  ->  dom  F  e.  _V )
76adantr 465 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  dom  F  e.  _V )
8 eleq1 2503 . . . . . . . . . 10  |-  ( A  =  dom  F  -> 
( A  e.  _V  <->  dom 
F  e.  _V )
)
98eqcoms 2446 . . . . . . . . 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 6710 . . . . . 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 1218 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
k  e.  ( F supp 
Z )  <->  ( k  e.  A  /\  ( F `  k )  =/=  Z ) ) )
16 eldif 3350 . . . . . . . . 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 2690 . . . . . 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 3374 . . 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 6705 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
27 0ss 3678 . . . 4  |-  (/)  C_  W
2826, 27syl6eqss 3418 . . 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 1369    e. wcel 1756    =/= wne 2618   _Vcvv 2984    \ cdif 3337    C_ wss 3340   (/)c0 3649   dom cdm 4852    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   supp csupp 6702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-supp 6703
This theorem is referenced by:  fsuppco2  7664  fsuppcor  7665  cantnfp1lem1  7898  cantnfp1lem3  7900  gsumzaddlem  16420  gsumzmhm  16442  gsum2d2lem  16477  lcomfsupp  16997  psrbaglesupp  17447  mplsubglem  17522  mpllsslem  17523  mplsubrglem  17529  mvrcl  17540  evlslem3  17612  frlmssuvc1  18231  frlmsslsp  18235  frlmup2  18239  deg1mul3le  21600  jensen  22394  resf1o  26042
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