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Theorem suppss 6922
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 5713 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43adantl 464 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  F  Fn  A )
5 fdm 5717 . . . . . . . 8  |-  ( F : A --> B  ->  dom  F  =  A )
6 dmexg 6704 . . . . . . . . . 10  |-  ( F  e.  _V  ->  dom  F  e.  _V )
76adantr 463 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  dom  F  e.  _V )
8 eleq1 2526 . . . . . . . . . 10  |-  ( A  =  dom  F  -> 
( A  e.  _V  <->  dom 
F  e.  _V )
)
98eqcoms 2466 . . . . . . . . 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 428 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  A  e.  _V )
13 simplr 753 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  Z  e.  _V )
14 elsuppfn 6899 . . . . . 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 1226 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
k  e.  ( F supp 
Z )  <->  ( k  e.  A  /\  ( F `  k )  =/=  Z ) ) )
16 eldif 3471 . . . . . . . . 9  |-  ( k  e.  ( A  \  W )  <->  ( k  e.  A  /\  -.  k  e.  W ) )
17 suppss.n . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
1817adantll 711 . . . . . . . . 9  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  Z )
1916, 18sylan2br 474 . . . . . . . 8  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  ( k  e.  A  /\  -.  k  e.  W
) )  ->  ( F `  k )  =  Z )
2019expr 613 . . . . . . 7  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  A
)  ->  ( -.  k  e.  W  ->  ( F `  k )  =  Z ) )
2120necon1ad 2670 . . . . . 6  |-  ( ( ( ( F  e. 
_V  /\  Z  e.  _V )  /\  ph )  /\  k  e.  A
)  ->  ( ( F `  k )  =/=  Z  ->  k  e.  W ) )
2221expimpd 601 . . . . 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 3495 . . 3  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F supp  Z )  C_  W
)
2524ex 432 . 2  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  ( ph  ->  ( F supp  Z )  C_  W
) )
26 supp0prc 6894 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
27 0ss 3813 . . . 4  |-  (/)  C_  W
2826, 27syl6eqss 3539 . . 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458    C_ wss 3461   (/)c0 3783   dom cdm 4988    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-supp 6892
This theorem is referenced by:  fsuppco2  7854  fsuppcor  7855  cantnfp1lem1  8088  cantnfp1lem3  8090  gsumzaddlem  17133  gsumzmhm  17155  gsum2d2lem  17197  lcomfsupp  17745  psrbaglesupp  18212  mplsubglem  18288  mpllsslem  18289  mplsubrglem  18295  mvrcl  18306  evlslem3  18378  frlmssuvc1  18996  frlmsslsp  18998  frlmup2  19001  deg1mul3le  22683  jensen  23516  resf1o  27784
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