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Theorem frnsuppeq 6913
Description: Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
Assertion
Ref Expression
frnsuppeq  |-  ( ( I  e.  V  /\  Z  e.  W )  ->  ( F : I --> S  ->  ( F supp  Z )  =  ( `' F " ( S 
\  { Z }
) ) ) )

Proof of Theorem frnsuppeq
StepHypRef Expression
1 fex 6133 . . . . . . 7  |-  ( ( F : I --> S  /\  I  e.  V )  ->  F  e.  _V )
21expcom 435 . . . . . 6  |-  ( I  e.  V  ->  ( F : I --> S  ->  F  e.  _V )
)
32adantr 465 . . . . 5  |-  ( ( I  e.  V  /\  Z  e.  W )  ->  ( F : I --> S  ->  F  e.  _V ) )
43imp 429 . . . 4  |-  ( ( ( I  e.  V  /\  Z  e.  W
)  /\  F :
I --> S )  ->  F  e.  _V )
5 simplr 754 . . . 4  |-  ( ( ( I  e.  V  /\  Z  e.  W
)  /\  F :
I --> S )  ->  Z  e.  W )
6 suppimacnv 6912 . . . 4  |-  ( ( F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
74, 5, 6syl2anc 661 . . 3  |-  ( ( ( I  e.  V  /\  Z  e.  W
)  /\  F :
I --> S )  -> 
( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
8 invdif 3739 . . . . . 6  |-  ( S  i^i  ( _V  \  { Z } ) )  =  ( S  \  { Z } )
98imaeq2i 5335 . . . . 5  |-  ( `' F " ( S  i^i  ( _V  \  { Z } ) ) )  =  ( `' F " ( S 
\  { Z }
) )
10 ffun 5733 . . . . . . 7  |-  ( F : I --> S  ->  Fun  F )
11 inpreima 6008 . . . . . . 7  |-  ( Fun 
F  ->  ( `' F " ( S  i^i  ( _V  \  { Z } ) ) )  =  ( ( `' F " S )  i^i  ( `' F " ( _V  \  { Z } ) ) ) )
1210, 11syl 16 . . . . . 6  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { Z }
) ) )  =  ( ( `' F " S )  i^i  ( `' F " ( _V 
\  { Z }
) ) ) )
13 cnvimass 5357 . . . . . . . 8  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
14 fdm 5735 . . . . . . . . 9  |-  ( F : I --> S  ->  dom  F  =  I )
15 fimacnv 6013 . . . . . . . . 9  |-  ( F : I --> S  -> 
( `' F " S )  =  I )
1614, 15eqtr4d 2511 . . . . . . . 8  |-  ( F : I --> S  ->  dom  F  =  ( `' F " S ) )
1713, 16syl5sseq 3552 . . . . . . 7  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { Z } ) )  C_  ( `' F " S ) )
18 sseqin2 3717 . . . . . . 7  |-  ( ( `' F " ( _V 
\  { Z }
) )  C_  ( `' F " S )  <-> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { Z }
) ) )  =  ( `' F "
( _V  \  { Z } ) ) )
1917, 18sylib 196 . . . . . 6  |-  ( F : I --> S  -> 
( ( `' F " S )  i^i  ( `' F " ( _V 
\  { Z }
) ) )  =  ( `' F "
( _V  \  { Z } ) ) )
2012, 19eqtrd 2508 . . . . 5  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { Z }
) ) )  =  ( `' F "
( _V  \  { Z } ) ) )
219, 20syl5reqr 2523 . . . 4  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { Z } ) )  =  ( `' F "
( S  \  { Z } ) ) )
2221adantl 466 . . 3  |-  ( ( ( I  e.  V  /\  Z  e.  W
)  /\  F :
I --> S )  -> 
( `' F "
( _V  \  { Z } ) )  =  ( `' F "
( S  \  { Z } ) ) )
237, 22eqtrd 2508 . 2  |-  ( ( ( I  e.  V  /\  Z  e.  W
)  /\  F :
I --> S )  -> 
( F supp  Z )  =  ( `' F " ( S  \  { Z } ) ) )
2423ex 434 1  |-  ( ( I  e.  V  /\  Z  e.  W )  ->  ( F : I --> S  ->  ( F supp  Z )  =  ( `' F " ( S 
\  { Z }
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   {csn 4027   `'ccnv 4998   dom cdm 4999   "cima 5002   Fun wfun 5582   -->wf 5584  (class class class)co 6284   supp csupp 6901
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
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-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  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-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-supp 6902
This theorem is referenced by:  frnfsuppbi  7858  frnnn0supp  10849  ffs2  27251  eulerpartlemmf  27982
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