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Theorem supp0cosupp0 6837
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 457 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  V )
21anim2i 569 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  F  e.  V ) )
32ancomd 451 . . . . . 6  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( F  e.  V  /\  Z  e.  _V )
)
4 suppimacnv 6810 . . . . . 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 6637 . . . . . . . . 9  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o.  G
)  e.  _V )
87anim2i 569 . . . . . . . 8  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  ( F  o.  G )  e.  _V ) )
98ancomd 451 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F  o.  G
)  e.  _V  /\  Z  e.  _V )
)
10 suppimacnv 6810 . . . . . . 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 5132 . . . . . . . . 9  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
1312imaeq1i 5273 . . . . . . . 8  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
14 imaco 5450 . . . . . . . 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 5272 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  ( `' G "
(/) ) )
17 ima0 5291 . . . . . . . 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 434 . . . 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 434 . 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 907 . . . . 5  |-  ( -.  Z  e.  _V  ->  -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )
)
26 supp0prc 6802 . . . . 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3076    \ cdif 3432   (/)c0 3744   {csn 3984   `'ccnv 4946   "cima 4950    o. ccom 4951  (class class class)co 6199   supp csupp 6799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-supp 6800
This theorem is referenced by:  gsumval3lem2  16504
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