Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ffsrn Structured version   Visualization version   Unicode version

Theorem ffsrn 28389
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 5254 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) )
21reseq2i 5108 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( ( `' F " ( _V 
\  { Z }
) )  u.  ( `' F " { Z } ) ) )
3 undif1 3833 . . . . . . . . 9  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  ( _V  u.  { Z } )
4 ssv 3438 . . . . . . . . . 10  |-  { Z }  C_  _V
5 ssequn2 3598 . . . . . . . . . 10  |-  ( { Z }  C_  _V  <->  ( _V  u.  { Z } )  =  _V )
64, 5mpbi 213 . . . . . . . . 9  |-  ( _V  u.  { Z }
)  =  _V
73, 6eqtri 2493 . . . . . . . 8  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  _V
87imaeq2i 5172 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( `' F " _V )
98reseq2i 5108 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( `' F " _V ) )
10 resundi 5124 . . . . . 6  |-  ( F  |`  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
112, 9, 103eqtr3i 2501 . . . . 5  |-  ( F  |`  ( `' F " _V ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
12 ffsrn.1 . . . . . 6  |-  ( ph  ->  Fun  F )
13 dfdm4 5032 . . . . . . 7  |-  dom  F  =  ran  `' F
14 dfrn4 5303 . . . . . . 7  |-  ran  `' F  =  ( `' F " _V )
1513, 14eqtri 2493 . . . . . 6  |-  dom  F  =  ( `' F " _V )
16 df-fn 5592 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  <->  ( Fun  F  /\  dom  F  =  ( `' F " _V ) ) )
17 fnresdm 5695 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  ->  ( F  |`  ( `' F " _V ) )  =  F )
1816, 17sylbir 218 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  ( `' F " _V ) )  -> 
( F  |`  ( `' F " _V )
)  =  F )
1912, 15, 18sylancl 675 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " _V )
)  =  F )
2011, 19syl5reqr 2520 . . . 4  |-  ( ph  ->  F  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
2120rneqd 5068 . . 3  |-  ( ph  ->  ran  F  =  ran  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
22 rnun 5250 . . 3  |-  ran  (
( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )  =  ( ran  ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )
2321, 22syl6eq 2521 . 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 6944 . . . . . 6  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
2724, 25, 26syl2anc 673 . . . . 5  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
28 ffsrn.2 . . . . 5  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
2927, 28eqeltrrd 2550 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
Fin )
30 cnvexg 6758 . . . . . 6  |-  ( F  e.  V  ->  `' F  e.  _V )
31 imaexg 6749 . . . . . 6  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  { Z }
) )  e.  _V )
3224, 30, 313syl 18 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
_V )
33 cnvimass 5194 . . . . . . 7  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
34 fores 5815 . . . . . . 7  |-  ( ( Fun  F  /\  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F )  ->  ( F  |`  ( `' F "
( _V  \  { Z } ) ) ) : ( `' F " ( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
3512, 33, 34sylancl 675 . . . . . 6  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) ) : ( `' F "
( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
36 fofn 5808 . . . . . 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 8981 . . . . 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 61 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) )
40 domfi 7811 . . . 4  |-  ( ( ( `' F "
( _V  \  { Z } ) )  e. 
Fin  /\  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  ~<_  ( `' F "
( _V  \  { Z } ) ) )  ->  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  e.  Fin )
4129, 39, 40syl2anc 673 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  e. 
Fin )
42 snfi 7668 . . . 4  |-  { Z }  e.  Fin
43 df-ima 4852 . . . . . 6  |-  ( F
" ( `' F " { Z } ) )  =  ran  ( F  |`  ( `' F " { Z } ) )
44 funimacnv 5665 . . . . . . 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 2519 . . . . 5  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
47 inss1 3643 . . . . 5  |-  ( { Z }  i^i  ran  F )  C_  { Z }
4846, 47syl6eqss 3468 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  C_  { Z } )
49 ssfi 7810 . . . 4  |-  ( ( { Z }  e.  Fin  /\  ran  ( F  |`  ( `' F " { Z } ) ) 
C_  { Z }
)  ->  ran  ( F  |`  ( `' F " { Z } ) )  e.  Fin )
5042, 48, 49sylancr 676 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  e. 
Fin )
51 unfi 7856 . . 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 673 . 2  |-  ( ph  ->  ( ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )  e.  Fin )
5323, 52eqeltrd 2549 1  |-  ( ph  ->  ran  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   {csn 3959   class class class wbr 4395   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583    Fn wfn 5584   -onto->wfo 5587  (class class class)co 6308   supp csupp 6933    ~<_ cdom 7585   Fincfn 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-ac2 8911
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-fin 7591  df-card 8391  df-acn 8394  df-ac 8565
This theorem is referenced by:  fpwrelmapffslem  28392
  Copyright terms: Public domain W3C validator