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Theorem frnsuppeq 6929
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 6146 . . . . . . 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 755 . . . 4  |-  ( ( ( I  e.  V  /\  Z  e.  W
)  /\  F :
I --> S )  ->  Z  e.  W )
6 suppimacnv 6928 . . . 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 3746 . . . . . 6  |-  ( S  i^i  ( _V  \  { Z } ) )  =  ( S  \  { Z } )
98imaeq2i 5345 . . . . 5  |-  ( `' F " ( S  i^i  ( _V  \  { Z } ) ) )  =  ( `' F " ( S 
\  { Z }
) )
10 ffun 5739 . . . . . . 7  |-  ( F : I --> S  ->  Fun  F )
11 inpreima 6015 . . . . . . 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 5367 . . . . . . . 8  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
14 fdm 5741 . . . . . . . . 9  |-  ( F : I --> S  ->  dom  F  =  I )
15 fimacnv 6020 . . . . . . . . 9  |-  ( F : I --> S  -> 
( `' F " S )  =  I )
1614, 15eqtr4d 2501 . . . . . . . 8  |-  ( F : I --> S  ->  dom  F  =  ( `' F " S ) )
1713, 16syl5sseq 3547 . . . . . . 7  |-  ( F : I --> S  -> 
( `' F "
( _V  \  { Z } ) )  C_  ( `' F " S ) )
18 sseqin2 3713 . . . . . . 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 2498 . . . . 5  |-  ( F : I --> S  -> 
( `' F "
( S  i^i  ( _V  \  { Z }
) ) )  =  ( `' F "
( _V  \  { Z } ) ) )
219, 20syl5reqr 2513 . . . 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 2498 . 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 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   `'ccnv 5007   dom cdm 5008   "cima 5011   Fun wfun 5588   -->wf 5590  (class class class)co 6296   supp csupp 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-supp 6918
This theorem is referenced by:  frnfsuppbi  7876  frnnn0supp  10870  ffs2  27701  eulerpartlemmf  28489
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