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Theorem ffsrn 28314
Description: The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
Hypotheses
Ref Expression
ffsrn.z  |-  ( ph  ->  Z  e.  W )
ffsrn.0  |-  ( ph  ->  F  e.  V )
ffsrn.1  |-  ( ph  ->  Fun  F )
ffsrn.2  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
Assertion
Ref Expression
ffsrn  |-  ( ph  ->  ran  F  e.  Fin )

Proof of Theorem ffsrn
StepHypRef Expression
1 imaundi 5248 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) )
21reseq2i 5102 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( ( `' F " ( _V 
\  { Z }
) )  u.  ( `' F " { Z } ) ) )
3 undif1 3842 . . . . . . . . 9  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  ( _V  u.  { Z } )
4 ssv 3452 . . . . . . . . . 10  |-  { Z }  C_  _V
5 ssequn2 3607 . . . . . . . . . 10  |-  ( { Z }  C_  _V  <->  ( _V  u.  { Z } )  =  _V )
64, 5mpbi 212 . . . . . . . . 9  |-  ( _V  u.  { Z }
)  =  _V
73, 6eqtri 2473 . . . . . . . 8  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  _V
87imaeq2i 5166 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( `' F " _V )
98reseq2i 5102 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( `' F " _V ) )
10 resundi 5118 . . . . . 6  |-  ( F  |`  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
112, 9, 103eqtr3i 2481 . . . . 5  |-  ( F  |`  ( `' F " _V ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
12 ffsrn.1 . . . . . 6  |-  ( ph  ->  Fun  F )
13 dfdm4 5027 . . . . . . 7  |-  dom  F  =  ran  `' F
14 dfrn4 5296 . . . . . . 7  |-  ran  `' F  =  ( `' F " _V )
1513, 14eqtri 2473 . . . . . 6  |-  dom  F  =  ( `' F " _V )
16 df-fn 5585 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  <->  ( Fun  F  /\  dom  F  =  ( `' F " _V ) ) )
17 fnresdm 5685 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  ->  ( F  |`  ( `' F " _V ) )  =  F )
1816, 17sylbir 217 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  ( `' F " _V ) )  -> 
( F  |`  ( `' F " _V )
)  =  F )
1912, 15, 18sylancl 668 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " _V )
)  =  F )
2011, 19syl5reqr 2500 . . . 4  |-  ( ph  ->  F  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
2120rneqd 5062 . . 3  |-  ( ph  ->  ran  F  =  ran  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
22 rnun 5244 . . 3  |-  ran  (
( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )  =  ( ran  ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )
2321, 22syl6eq 2501 . 2  |-  ( ph  ->  ran  F  =  ( ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u. 
ran  ( F  |`  ( `' F " { Z } ) ) ) )
24 ffsrn.0 . . . . . 6  |-  ( ph  ->  F  e.  V )
25 ffsrn.z . . . . . 6  |-  ( ph  ->  Z  e.  W )
26 suppimacnv 6925 . . . . . 6  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
2724, 25, 26syl2anc 667 . . . . 5  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
28 ffsrn.2 . . . . 5  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
2927, 28eqeltrrd 2530 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
Fin )
30 cnvexg 6739 . . . . . 6  |-  ( F  e.  V  ->  `' F  e.  _V )
31 imaexg 6730 . . . . . 6  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  { Z }
) )  e.  _V )
3224, 30, 313syl 18 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
_V )
33 cnvimass 5188 . . . . . . 7  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
34 fores 5802 . . . . . . 7  |-  ( ( Fun  F  /\  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F )  ->  ( F  |`  ( `' F "
( _V  \  { Z } ) ) ) : ( `' F " ( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
3512, 33, 34sylancl 668 . . . . . 6  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) ) : ( `' F "
( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
36 fofn 5795 . . . . . 6  |-  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) ) : ( `' F " ( _V 
\  { Z }
) ) -onto-> ( F
" ( `' F " ( _V  \  { Z } ) ) )  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) ) )
3735, 36syl 17 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) ) )
38 fnrndomg 8963 . . . . 5  |-  ( ( `' F " ( _V 
\  { Z }
) )  e.  _V  ->  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) )  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) ) )
3932, 37, 38sylc 62 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) )
40 domfi 7793 . . . 4  |-  ( ( ( `' F "
( _V  \  { Z } ) )  e. 
Fin  /\  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  ~<_  ( `' F "
( _V  \  { Z } ) ) )  ->  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  e.  Fin )
4129, 39, 40syl2anc 667 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  e. 
Fin )
42 snfi 7650 . . . 4  |-  { Z }  e.  Fin
43 df-ima 4847 . . . . . 6  |-  ( F
" ( `' F " { Z } ) )  =  ran  ( F  |`  ( `' F " { Z } ) )
44 funimacnv 5655 . . . . . . 7  |-  ( Fun 
F  ->  ( F " ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F )
)
4512, 44syl 17 . . . . . 6  |-  ( ph  ->  ( F " ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
4643, 45syl5eqr 2499 . . . . 5  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
47 inss1 3652 . . . . 5  |-  ( { Z }  i^i  ran  F )  C_  { Z }
4846, 47syl6eqss 3482 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  C_  { Z } )
49 ssfi 7792 . . . 4  |-  ( ( { Z }  e.  Fin  /\  ran  ( F  |`  ( `' F " { Z } ) ) 
C_  { Z }
)  ->  ran  ( F  |`  ( `' F " { Z } ) )  e.  Fin )
5042, 48, 49sylancr 669 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  e. 
Fin )
51 unfi 7838 . . 3  |-  ( ( ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  e. 
Fin  /\  ran  ( F  |`  ( `' F " { Z } ) )  e.  Fin )  -> 
( ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )  e.  Fin )
5241, 50, 51syl2anc 667 . 2  |-  ( ph  ->  ( ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )  e.  Fin )
5323, 52eqeltrd 2529 1  |-  ( ph  ->  ran  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   {csn 3968   class class class wbr 4402   `'ccnv 4833   dom cdm 4834   ran crn 4835    |` cres 4836   "cima 4837   Fun wfun 5576    Fn wfn 5577   -onto->wfo 5580  (class class class)co 6290   supp csupp 6914    ~<_ cdom 7567   Fincfn 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-ac2 8893
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-fin 7573  df-card 8373  df-acn 8376  df-ac 8547
This theorem is referenced by:  fpwrelmapffslem  28317
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