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Theorem supp0cosupp0 6936
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 6909 . . . . . 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 2469 . . . 4  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F supp  Z )  =  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) ) )
7 coexg 6732 . . . . . . . . 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 6909 . . . . . . 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 5186 . . . . . . . . 9  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
1312imaeq1i 5332 . . . . . . . 8  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
14 imaco 5510 . . . . . . . 8  |-  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
1513, 14eqtri 2496 . . . . . . 7  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
16 imaeq2 5331 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  ( `' G "
(/) ) )
17 ima0 5350 . . . . . . . 8  |-  ( `' G " (/) )  =  (/)
1816, 17syl6eq 2524 . . . . . . 7  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  (/) )
1915, 18syl5eq 2520 . . . . . 6  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' ( F  o.  G ) "
( _V  \  { Z } ) )  =  (/) )
2011, 19sylan9eq 2528 . . . . 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 914 . . . . 5  |-  ( -.  Z  e.  _V  ->  -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )
)
26 supp0prc 6901 . . . . 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 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473   (/)c0 3785   {csn 4027   `'ccnv 4998   "cima 5002    o. ccom 5003  (class class class)co 6282   supp csupp 6898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-supp 6899
This theorem is referenced by:  gsumval3lem2  16701
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