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Theorem ffsrn 27709
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 5425 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) )
21reseq2i 5280 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( ( `' F " ( _V 
\  { Z }
) )  u.  ( `' F " { Z } ) ) )
3 undif1 3906 . . . . . . . . 9  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  ( _V  u.  { Z } )
4 ssv 3519 . . . . . . . . . 10  |-  { Z }  C_  _V
5 ssequn2 3673 . . . . . . . . . 10  |-  ( { Z }  C_  _V  <->  ( _V  u.  { Z } )  =  _V )
64, 5mpbi 208 . . . . . . . . 9  |-  ( _V  u.  { Z }
)  =  _V
73, 6eqtri 2486 . . . . . . . 8  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  _V
87imaeq2i 5345 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( `' F " _V )
98reseq2i 5280 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( `' F " _V ) )
10 resundi 5297 . . . . . 6  |-  ( F  |`  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
112, 9, 103eqtr3i 2494 . . . . 5  |-  ( F  |`  ( `' F " _V ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
12 ffsrn.1 . . . . . 6  |-  ( ph  ->  Fun  F )
13 dfdm4 5205 . . . . . . 7  |-  dom  F  =  ran  `' F
14 dfrn4 5473 . . . . . . 7  |-  ran  `' F  =  ( `' F " _V )
1513, 14eqtri 2486 . . . . . 6  |-  dom  F  =  ( `' F " _V )
16 df-fn 5597 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  <->  ( Fun  F  /\  dom  F  =  ( `' F " _V ) ) )
17 fnresdm 5696 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  ->  ( F  |`  ( `' F " _V ) )  =  F )
1816, 17sylbir 213 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  ( `' F " _V ) )  -> 
( F  |`  ( `' F " _V )
)  =  F )
1912, 15, 18sylancl 662 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " _V )
)  =  F )
2011, 19syl5reqr 2513 . . . 4  |-  ( ph  ->  F  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
2120rneqd 5240 . . 3  |-  ( ph  ->  ran  F  =  ran  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
22 rnun 5421 . . 3  |-  ran  (
( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )  =  ( ran  ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )
2321, 22syl6eq 2514 . 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 6928 . . . . . 6  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
2724, 25, 26syl2anc 661 . . . . 5  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
28 ffsrn.2 . . . . 5  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
2927, 28eqeltrrd 2546 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
Fin )
30 cnvexg 6745 . . . . . 6  |-  ( F  e.  V  ->  `' F  e.  _V )
31 imaexg 6736 . . . . . 6  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  { Z }
) )  e.  _V )
3224, 30, 313syl 20 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
_V )
33 cnvimass 5367 . . . . . . 7  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
34 fores 5810 . . . . . . 7  |-  ( ( Fun  F  /\  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F )  ->  ( F  |`  ( `' F "
( _V  \  { Z } ) ) ) : ( `' F " ( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
3512, 33, 34sylancl 662 . . . . . 6  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) ) : ( `' F "
( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
36 fofn 5803 . . . . . 6  |-  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) ) : ( `' F " ( _V 
\  { Z }
) ) -onto-> ( F
" ( `' F " ( _V  \  { Z } ) ) )  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) ) )
3735, 36syl 16 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) ) )
38 fnrndomg 8930 . . . . 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 60 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) )
40 domfi 7760 . . . 4  |-  ( ( ( `' F "
( _V  \  { Z } ) )  e. 
Fin  /\  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  ~<_  ( `' F "
( _V  \  { Z } ) ) )  ->  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  e.  Fin )
4129, 39, 40syl2anc 661 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  e. 
Fin )
42 snfi 7615 . . . 4  |-  { Z }  e.  Fin
43 df-ima 5021 . . . . . 6  |-  ( F
" ( `' F " { Z } ) )  =  ran  ( F  |`  ( `' F " { Z } ) )
44 funimacnv 5666 . . . . . . 7  |-  ( Fun 
F  ->  ( F " ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F )
)
4512, 44syl 16 . . . . . 6  |-  ( ph  ->  ( F " ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
4643, 45syl5eqr 2512 . . . . 5  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
47 inss1 3714 . . . . 5  |-  ( { Z }  i^i  ran  F )  C_  { Z }
4846, 47syl6eqss 3549 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  C_  { Z } )
49 ssfi 7759 . . . 4  |-  ( ( { Z }  e.  Fin  /\  ran  ( F  |`  ( `' F " { Z } ) ) 
C_  { Z }
)  ->  ran  ( F  |`  ( `' F " { Z } ) )  e.  Fin )
5042, 48, 49sylancr 663 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  e. 
Fin )
51 unfi 7805 . . 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 661 . 2  |-  ( ph  ->  ( ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )  e.  Fin )
5323, 52eqeltrd 2545 1  |-  ( ph  ->  ran  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   {csn 4032   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Fun wfun 5588    Fn wfn 5589   -onto->wfo 5592  (class class class)co 6296   supp csupp 6917    ~<_ cdom 7533   Fincfn 7535
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-pow 4634  ax-pr 4695  ax-un 6591  ax-ac2 8860
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-rmo 2815  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-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  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-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-fin 7539  df-card 8337  df-acn 8340  df-ac 8514
This theorem is referenced by:  fpwrelmapffslem  27712
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