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Theorem imacosupp 6941
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 5188 . . . . . . . 8  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
21imaeq1i 5334 . . . . . . 7  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
3 imaco 5512 . . . . . . 7  |-  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
42, 3eqtri 2496 . . . . . 6  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
54imaeq2i 5335 . . . . 5  |-  ( G
" ( `' ( F  o.  G )
" ( _V  \  { Z } ) ) )  =  ( G
" ( `' G " ( `' F "
( _V  \  { Z } ) ) ) )
6 funforn 5802 . . . . . . . 8  |-  ( Fun 
G  <->  G : dom  G -onto-> ran  G )
76biimpi 194 . . . . . . 7  |-  ( Fun 
G  ->  G : dom  G -onto-> ran  G )
87ad2antrl 727 . . . . . 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 457 . . . . . . . . . . . . 13  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  V )
109anim2i 569 . . . . . . . . . . . 12  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  F  e.  V ) )
1110ancomd 451 . . . . . . . . . . 11  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( F  e.  V  /\  Z  e.  _V )
)
12 suppimacnv 6913 . . . . . . . . . . 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 3531 . . . . . . . . 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 467 . . . . . . 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 429 . . . . . 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 5833 . . . . . 6  |-  ( ( G : dom  G -onto-> ran  G  /\  ( `' F " ( _V 
\  { Z }
) )  C_  ran  G )  ->  ( G " ( `' G "
( `' F "
( _V  \  { Z } ) ) ) )  =  ( `' F " ( _V 
\  { Z }
) ) )
198, 17, 18syl2anc 661 . . . . 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 2520 . . . 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 6736 . . . . . . . . 9  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o.  G
)  e.  _V )
2221anim2i 569 . . . . . . . 8  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( Z  e.  _V  /\  ( F  o.  G )  e.  _V ) )
2322ancomd 451 . . . . . . 7  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  (
( F  o.  G
)  e.  _V  /\  Z  e.  _V )
)
24 suppimacnv 6913 . . . . . . 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 5337 . . . . 5  |-  ( ( Z  e.  _V  /\  ( F  e.  V  /\  G  e.  W
) )  ->  ( G " ( ( F  o.  G ) supp  Z
) )  =  ( G " ( `' ( F  o.  G
) " ( _V 
\  { Z }
) ) ) )
2726adantr 465 . . . 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 465 . . . 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 2518 . . 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 604 . 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 5352 . . . . 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 6905 . . . . . . 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 5337 . . . . 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 6905 . . . . . 6  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
3937, 38syl 16 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F supp  Z )  =  (/) )
4031, 36, 393eqtr4a 2534 . . . 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    o. ccom 5003   Fun wfun 5582   -onto->wfo 5586  (class class class)co 6285   supp csupp 6902
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 6577
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-supp 6903
This theorem is referenced by:  gsumval3lem1  16724  gsumval3lem2  16725
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