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Theorem imacosupp 6932
Description: The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
Assertion
Ref Expression
imacosupp  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( Fun  G  /\  ( F supp  Z ) 
C_  ran  G )  ->  ( G " (
( F  o.  G
) supp  Z ) )  =  ( F supp  Z ) ) )

Proof of Theorem imacosupp
StepHypRef Expression
1 cnvco 5177 . . . . . . . 8  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
21imaeq1i 5322 . . . . . . 7  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
3 imaco 5495 . . . . . . 7  |-  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
42, 3eqtri 2483 . . . . . 6  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
54imaeq2i 5323 . . . . 5  |-  ( G
" ( `' ( F  o.  G )
" ( _V  \  { Z } ) ) )  =  ( G
" ( `' G " ( `' F "
( _V  \  { Z } ) ) ) )
6 funforn 5784 . . . . . . . 8  |-  ( Fun 
G  <->  G : dom  G -onto-> ran  G )
76biimpi 194 . . . . . . 7  |-  ( Fun 
G  ->  G : dom  G -onto-> ran  G )
87ad2antrl 725 . . . . . 6  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  G : dom  G -onto-> ran  G
)
9 simpl 455 . . . . . . . . . . . . 13  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  V )
109anim2i 567 . . . . . . . . . . . 12  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  F  e.  V ) )
1110ancomd 449 . . . . . . . . . . 11  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( F  e.  V  /\  Z  e.  _V )
)
12 suppimacnv 6902 . . . . . . . . . . 11  |-  ( ( F  e.  V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( F supp  Z )  =  ( `' F " ( _V 
\  { Z }
) ) )
1413sseq1d 3516 . . . . . . . . 9  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F supp  Z )  C_ 
ran  G  <->  ( `' F " ( _V  \  { Z } ) )  C_  ran  G ) )
1514biimpd 207 . . . . . . . 8  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F supp  Z )  C_ 
ran  G  ->  ( `' F " ( _V 
\  { Z }
) )  C_  ran  G ) )
1615adantld 465 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( Fun  G  /\  ( F supp  Z )  C_ 
ran  G )  -> 
( `' F "
( _V  \  { Z } ) )  C_  ran  G ) )
1716imp 427 . . . . . 6  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  ( `' F " ( _V 
\  { Z }
) )  C_  ran  G )
18 foimacnv 5815 . . . . . 6  |-  ( ( G : dom  G -onto-> ran  G  /\  ( `' F " ( _V 
\  { Z }
) )  C_  ran  G )  ->  ( G " ( `' G "
( `' F "
( _V  \  { Z } ) ) ) )  =  ( `' F " ( _V 
\  { Z }
) ) )
198, 17, 18syl2anc 659 . . . . 5  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  ( G " ( `' G " ( `' F "
( _V  \  { Z } ) ) ) )  =  ( `' F " ( _V 
\  { Z }
) ) )
205, 19syl5eq 2507 . . . 4  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  ( G " ( `' ( F  o.  G )
" ( _V  \  { Z } ) ) )  =  ( `' F " ( _V 
\  { Z }
) ) )
21 coexg 6724 . . . . . . . . 9  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o.  G
)  e.  _V )
2221anim2i 567 . . . . . . . 8  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  ( F  o.  G )  e.  _V ) )
2322ancomd 449 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F  o.  G
)  e.  _V  /\  Z  e.  _V )
)
24 suppimacnv 6902 . . . . . . 7  |-  ( ( ( F  o.  G
)  e.  _V  /\  Z  e.  _V )  ->  ( ( F  o.  G ) supp  Z )  =  ( `' ( F  o.  G )
" ( _V  \  { Z } ) ) )
2523, 24syl 16 . . . . . 6  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F  o.  G
) supp  Z )  =  ( `' ( F  o.  G ) " ( _V  \  { Z }
) ) )
2625imaeq2d 5325 . . . . 5  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( G " ( ( F  o.  G ) supp  Z
) )  =  ( G " ( `' ( F  o.  G
) " ( _V 
\  { Z }
) ) ) )
2726adantr 463 . . . 4  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  ( G " ( ( F  o.  G ) supp  Z
) )  =  ( G " ( `' ( F  o.  G
) " ( _V 
\  { Z }
) ) ) )
2813adantr 463 . . . 4  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  ( F supp  Z )  =  ( `' F " ( _V 
\  { Z }
) ) )
2920, 27, 283eqtr4d 2505 . . 3  |-  ( ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  /\  ( Fun  G  /\  ( F supp 
Z )  C_  ran  G ) )  ->  ( G " ( ( F  o.  G ) supp  Z
) )  =  ( F supp  Z ) )
3029exp31 602 . 2  |-  ( Z  e.  _V  ->  (
( F  e.  V  /\  G  e.  W
)  ->  ( ( Fun  G  /\  ( F supp 
Z )  C_  ran  G )  ->  ( G " ( ( F  o.  G ) supp  Z )
)  =  ( F supp 
Z ) ) ) )
31 ima0 5340 . . . . 5  |-  ( G
" (/) )  =  (/)
32 id 22 . . . . . . . 8  |-  ( -.  Z  e.  _V  ->  -.  Z  e.  _V )
3332intnand 914 . . . . . . 7  |-  ( -.  Z  e.  _V  ->  -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )
)
34 supp0prc 6894 . . . . . . 7  |-  ( -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )  ->  ( ( F  o.  G ) supp  Z )  =  (/) )
3533, 34syl 16 . . . . . 6  |-  ( -.  Z  e.  _V  ->  ( ( F  o.  G
) supp  Z )  =  (/) )
3635imaeq2d 5325 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( G " ( ( F  o.  G ) supp 
Z ) )  =  ( G " (/) ) )
3732intnand 914 . . . . . 6  |-  ( -.  Z  e.  _V  ->  -.  ( F  e.  _V  /\  Z  e.  _V )
)
38 supp0prc 6894 . . . . . 6  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
3937, 38syl 16 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F supp  Z )  =  (/) )
4031, 36, 393eqtr4a 2521 . . . 4  |-  ( -.  Z  e.  _V  ->  ( G " ( ( F  o.  G ) supp 
Z ) )  =  ( F supp  Z ) )
4140a1d 25 . . 3  |-  ( -.  Z  e.  _V  ->  ( ( Fun  G  /\  ( F supp  Z )  C_ 
ran  G )  -> 
( G " (
( F  o.  G
) supp  Z ) )  =  ( F supp  Z ) ) )
4241a1d 25 . 2  |-  ( -.  Z  e.  _V  ->  ( ( F  e.  V  /\  G  e.  W
)  ->  ( ( Fun  G  /\  ( F supp 
Z )  C_  ran  G )  ->  ( G " ( ( F  o.  G ) supp  Z )
)  =  ( F supp 
Z ) ) ) )
4330, 42pm2.61i 164 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( Fun  G  /\  ( F supp  Z ) 
C_  ran  G )  ->  ( G " (
( F  o.  G
) supp  Z ) )  =  ( F 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    C_ wss 3461   (/)c0 3783   {csn 4016   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    o. ccom 4992   Fun wfun 5564   -onto->wfo 5568  (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-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-supp 6892
This theorem is referenced by:  gsumval3lem1  17108  gsumval3lem2  17109
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