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Theorem suppss 6718
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 5556 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43adantl 463 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  F  Fn  A )
5 fdm 5560 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
6 dmexg 6508 . . . . . . . . . 10  |-  ( F  e.  _V  ->  dom  F  e.  _V )
76adantr 462 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  dom  F  e.  _V )
8 eleq1 2501 . . . . . . . . . 10  |-  ( A  =  dom  F  -> 
( A  e.  _V  <->  dom 
F  e.  _V )
)
98eqcoms 2444 . . . . . . . . 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 749 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  Z  e.  _V )
14 elsuppfn 6697 . . . . . 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 1213 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
k  e.  ( F supp 
Z )  <->  ( k  e.  A  /\  ( F `  k )  =/=  Z ) ) )
16 eldif 3335 . . . . . . . . 9  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
17 suppss.n . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
1817adantll 708 . . . . . . . . 9  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
1916, 18sylan2br 473 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  ( k  e.  A  /\  -.  k  e.  W
) )  ->  ( F `  k )  =  Z )
2019expr 612 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  A
)  ->  ( -.  k  e.  W  ->  ( F `  k )  =  Z ) )
2120necon1ad 2676 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  A
)  ->  ( ( F `  k )  =/=  Z  ->  k  e.  W ) )
2221expimpd 600 . . . . 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 3359 . . 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 6692 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
27 0ss 3663 . . . 4  |-  (/)  C_  W
2826, 27syl6eqss 3403 . . 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    \ cdif 3322    C_ wss 3325   (/)c0 3634   dom cdm 4836    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   supp csupp 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-supp 6690
This theorem is referenced by:  fsuppco2  7648  fsuppcor  7649  cantnfp1lem1  7882  cantnfp1lem3  7884  gsumzaddlem  16401  gsumzmhm  16422  gsum2d2lem  16455  lcomfsupp  16965  psrbaglesupp  17425  mplsubglem  17500  mpllsslem  17501  mplsubrglem  17507  mvrcl  17518  frlmssuvc1  18119  frlmsslsp  18123  frlmup2  18127  evlslem3  21424  deg1mul3le  21531  jensen  22325  resf1o  25949
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