MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supp0cosupp0 Structured version   Unicode version

Theorem supp0cosupp0 6941
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 6914 . . . . . 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 2445 . . . 4  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F supp  Z )  =  (/)  <->  ( `' F " ( _V  \  { Z } ) )  =  (/) ) )
7 coexg 6736 . . . . . . . . 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 6914 . . . . . . 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 5178 . . . . . . . . 9  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
1312imaeq1i 5324 . . . . . . . 8  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
14 imaco 5502 . . . . . . . 8  |-  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
1513, 14eqtri 2472 . . . . . . 7  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
16 imaeq2 5323 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  ( `' G "
(/) ) )
17 ima0 5342 . . . . . . . 8  |-  ( `' G " (/) )  =  (/)
1816, 17syl6eq 2500 . . . . . . 7  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' G "
( `' F "
( _V  \  { Z } ) ) )  =  (/) )
1915, 18syl5eq 2496 . . . . . 6  |-  ( ( `' F " ( _V 
\  { Z }
) )  =  (/)  ->  ( `' ( F  o.  G ) "
( _V  \  { Z } ) )  =  (/) )
2011, 19sylan9eq 2504 . . . . 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 916 . . . . 5  |-  ( -.  Z  e.  _V  ->  -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )
)
26 supp0prc 6906 . . . . 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 1383    e. wcel 1804   _Vcvv 3095    \ cdif 3458   (/)c0 3770   {csn 4014   `'ccnv 4988   "cima 4992    o. ccom 4993  (class class class)co 6281   supp csupp 6903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-supp 6904
This theorem is referenced by:  gsumval3lem2  16888
  Copyright terms: Public domain W3C validator