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Theorem imacosupp 6741
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 5037 . . . . . . . 8  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
21imaeq1i 5178 . . . . . . 7  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )
3 imaco 5355 . . . . . . 7  |-  ( ( `' G  o.  `' F ) " ( _V  \  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
42, 3eqtri 2463 . . . . . 6  |-  ( `' ( F  o.  G
) " ( _V 
\  { Z }
) )  =  ( `' G " ( `' F " ( _V 
\  { Z }
) ) )
54imaeq2i 5179 . . . . 5  |-  ( G
" ( `' ( F  o.  G )
" ( _V  \  { Z } ) ) )  =  ( G
" ( `' G " ( `' F "
( _V  \  { Z } ) ) ) )
6 funforn 5639 . . . . . . . 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 6713 . . . . . . . . . . 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 3395 . . . . . . . . 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 5670 . . . . . 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 2487 . . . 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 6540 . . . . . . . . 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 6713 . . . . . . 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 5181 . . . . 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 2485 . . 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 5196 . . . . 5  |-  ( G
" (/) )  =  (/)
32 id 22 . . . . . . . 8  |-  ( -.  Z  e.  _V  ->  -.  Z  e.  _V )
3332intnand 907 . . . . . . 7  |-  ( -.  Z  e.  _V  ->  -.  ( ( F  o.  G )  e.  _V  /\  Z  e.  _V )
)
34 supp0prc 6705 . . . . . . 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 5181 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( G " ( ( F  o.  G ) supp 
Z ) )  =  ( G " (/) ) )
3732intnand 907 . . . . . 6  |-  ( -.  Z  e.  _V  ->  -.  ( F  e.  _V  /\  Z  e.  _V )
)
38 supp0prc 6705 . . . . . 6  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
3937, 38syl 16 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F supp  Z )  =  (/) )
4031, 36, 393eqtr4a 2501 . . . 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 1369    e. wcel 1756   _Vcvv 2984    \ cdif 3337    C_ wss 3340   (/)c0 3649   {csn 3889   `'ccnv 4851   dom cdm 4852   ran crn 4853   "cima 4855    o. ccom 4856   Fun wfun 5424   -onto->wfo 5428  (class class class)co 6103   supp csupp 6702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-supp 6703
This theorem is referenced by:  gsumval3lem1  16395  gsumval3lem2  16396
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