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Theorem supp0cosupp0 6931
Description: The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
Assertion
Ref Expression
supp0cosupp0  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F supp  Z
)  =  (/)  ->  (
( F  o.  G
) supp  Z )  =  (/) ) )

Proof of Theorem supp0cosupp0
StepHypRef Expression
1 simpl 455 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  V )
21anim2i 567 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  F  e.  V ) )
32ancomd 449 . . . . . 6  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( F  e.  V  /\  Z  e.  _V )
)
4 suppimacnv 6902 . . . . . 6  |-  ( ( F  e.  V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
53, 4syl 16 . . . . 5  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( F supp  Z )  =  ( `' F " ( _V 
\  { Z }
) ) )
65eqeq1d 2456 . . . 4  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F supp  Z )  =  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) ) )
7 coexg 6724 . . . . . . . . 9  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o.  G
)  e.  _V )
87anim2i 567 . . . . . . . 8  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  ( F  o.  G )  e.  _V ) )
98ancomd 449 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F  o.  G
)  e.  _V  /\  Z  e.  _V )
)
10 suppimacnv 6902 . . . . . . 7  |-  ( ( ( F  o.  G
)  e.  _V  /\  Z  e.  _V )  ->  ( ( F  o.  G ) supp  Z )  =  ( `' ( F  o.  G )
" ( _V  \  { Z } ) ) )
119, 10syl 16 . . . . . 6  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F  o.  G
) supp  Z )  =  ( `' ( F  o.  G ) " ( _V  \  { Z }
) ) )
12 cnvco 5177 . . . . . . . . 9  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
1312imaeq1i 5322 . . . . . . . 8  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
14 imaco 5495 . . . . . . . 8  |-  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
1513, 14eqtri 2483 . . . . . . 7  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
16 imaeq2 5321 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  ( `' G "
(/) ) )
17 ima0 5340 . . . . . . . 8  |-  ( `' G " (/) )  =  (/)
1816, 17syl6eq 2511 . . . . . . 7  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  (/) )
1915, 18syl5eq 2507 . . . . . 6  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' ( F  o.  G ) "
( _V  \  { Z } ) )  =  (/) )
2011, 19sylan9eq 2515 . . . . 5  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( `' F " ( _V 
\  { Z }
) )  =  (/) )  ->  ( ( F  o.  G ) supp  Z
)  =  (/) )
2120ex 432 . . . 4  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( `' F "
( _V  \  { Z } ) )  =  (/)  ->  ( ( F  o.  G ) supp  Z
)  =  (/) ) )
226, 21sylbid 215 . . 3  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F supp  Z )  =  (/)  ->  ( ( F  o.  G ) supp  Z )  =  (/) ) )
2322ex 432 . 2  |-  ( Z  e.  _V  ->  (
( F  e.  V  /\  G  e.  W
)  ->  ( ( F supp  Z )  =  (/)  ->  ( ( F  o.  G ) supp  Z )  =  (/) ) ) )
24 id 22 . . . . . 6  |-  ( -.  Z  e.  _V  ->  -.  Z  e.  _V )
2524intnand 914 . . . . 5  |-  ( -.  Z  e.  _V  ->  -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )
)
26 supp0prc 6894 . . . . 5  |-  ( -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )  ->  ( ( F  o.  G ) supp  Z )  =  (/) )
2725, 26syl 16 . . . 4  |-  ( -.  Z  e.  _V  ->  ( ( F  o.  G
) supp  Z )  =  (/) )
2827a1d 25 . . 3  |-  ( -.  Z  e.  _V  ->  ( ( F supp  Z )  =  (/)  ->  ( ( F  o.  G ) supp 
Z )  =  (/) ) )
2928a1d 25 . 2  |-  ( -.  Z  e.  _V  ->  ( ( F  e.  V  /\  G  e.  W
)  ->  ( ( F supp  Z )  =  (/)  ->  ( ( F  o.  G ) supp  Z )  =  (/) ) ) )
3023, 29pm2.61i 164 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F supp  Z
)  =  (/)  ->  (
( F  o.  G
) supp  Z )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458   (/)c0 3783   {csn 4016   `'ccnv 4987   "cima 4991    o. ccom 4992  (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-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-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  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-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-supp 6892
This theorem is referenced by:  gsumval3lem2  17112
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