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Theorem ffsrn 27074
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 undif1 3895 . . . . . . . . 9  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  ( _V  u.  { Z } )
2 ssv 3517 . . . . . . . . . 10  |-  { Z }  C_  _V
3 ssequn2 3670 . . . . . . . . . 10  |-  ( { Z }  C_  _V  <->  ( _V  u.  { Z } )  =  _V )
42, 3mpbi 208 . . . . . . . . 9  |-  ( _V  u.  { Z }
)  =  _V
51, 4eqtri 2489 . . . . . . . 8  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  _V
65imaeq2i 5326 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( `' F " _V )
76reseq2i 5261 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( `' F " _V ) )
8 imaundi 5409 . . . . . . . 8  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) )
98reseq2i 5261 . . . . . . 7  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( ( `' F " ( _V 
\  { Z }
) )  u.  ( `' F " { Z } ) ) )
10 resundi 5278 . . . . . . 7  |-  ( F  |`  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
119, 10eqtri 2489 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
127, 11eqtr3i 2491 . . . . 5  |-  ( F  |`  ( `' F " _V ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
13 ffsrn.1 . . . . . 6  |-  ( ph  ->  Fun  F )
14 dfdm4 5186 . . . . . . 7  |-  dom  F  =  ran  `' F
15 dfrn4 5458 . . . . . . 7  |-  ran  `' F  =  ( `' F " _V )
1614, 15eqtri 2489 . . . . . 6  |-  dom  F  =  ( `' F " _V )
17 df-fn 5582 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  <->  ( Fun  F  /\  dom  F  =  ( `' F " _V ) ) )
18 fnresdm 5681 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  ->  ( F  |`  ( `' F " _V ) )  =  F )
1917, 18sylbir 213 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  ( `' F " _V ) )  -> 
( F  |`  ( `' F " _V )
)  =  F )
2013, 16, 19sylancl 662 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " _V )
)  =  F )
2112, 20syl5reqr 2516 . . . 4  |-  ( ph  ->  F  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
2221rneqd 5221 . . 3  |-  ( ph  ->  ran  F  =  ran  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
23 rnun 5405 . . . 4  |-  ran  (
( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )  =  ( ran  ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )
2423a1i 11 . . 3  |-  ( ph  ->  ran  ( ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )  =  ( ran  ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) ) )
2522, 24eqtrd 2501 . 2  |-  ( ph  ->  ran  F  =  ( ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u. 
ran  ( F  |`  ( `' F " { Z } ) ) ) )
26 ffsrn.0 . . . . . 6  |-  ( ph  ->  F  e.  V )
27 ffsrn.z . . . . . 6  |-  ( ph  ->  Z  e.  W )
28 suppimacnv 6902 . . . . . 6  |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
2926, 27, 28syl2anc 661 . . . . 5  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
30 ffsrn.2 . . . . 5  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
3129, 30eqeltrrd 2549 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
Fin )
32 cnvimass 5348 . . . . . . 7  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
33 fores 5795 . . . . . . 7  |-  ( ( Fun  F  /\  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F )  ->  ( F  |`  ( `' F "
( _V  \  { Z } ) ) ) : ( `' F " ( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
3413, 32, 33sylancl 662 . . . . . 6  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) ) : ( `' F "
( _V  \  { Z } ) ) -onto-> ( F " ( `' F " ( _V 
\  { Z }
) ) ) )
35 fofn 5788 . . . . . 6  |-  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) ) : ( `' F " ( _V 
\  { Z }
) ) -onto-> ( F
" ( `' F " ( _V  \  { Z } ) ) )  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) ) )
3634, 35syl 16 . . . . 5  |-  ( ph  ->  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) ) )
37 cnvexg 6720 . . . . . . 7  |-  ( F  e.  V  ->  `' F  e.  _V )
38 imaexg 6711 . . . . . . 7  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  { Z }
) )  e.  _V )
3926, 37, 383syl 20 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
_V )
40 fnrndomg 8902 . . . . . 6  |-  ( ( `' F " ( _V 
\  { Z }
) )  e.  _V  ->  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) )  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) ) )
4139, 40syl 16 . . . . 5  |-  ( ph  ->  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  Fn  ( `' F "
( _V  \  { Z } ) )  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) ) )
4236, 41mpd 15 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  ~<_  ( `' F " ( _V 
\  { Z }
) ) )
43 domfi 7731 . . . 4  |-  ( ( ( `' F "
( _V  \  { Z } ) )  e. 
Fin  /\  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  ~<_  ( `' F "
( _V  \  { Z } ) ) )  ->  ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  e.  Fin )
4431, 42, 43syl2anc 661 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  e. 
Fin )
45 snfi 7586 . . . 4  |-  { Z }  e.  Fin
46 df-ima 5005 . . . . . 6  |-  ( F
" ( `' F " { Z } ) )  =  ran  ( F  |`  ( `' F " { Z } ) )
47 funimacnv 5651 . . . . . . 7  |-  ( Fun 
F  ->  ( F " ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F )
)
4813, 47syl 16 . . . . . 6  |-  ( ph  ->  ( F " ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
4946, 48syl5eqr 2515 . . . . 5  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
50 inss1 3711 . . . . 5  |-  ( { Z }  i^i  ran  F )  C_  { Z }
5149, 50syl6eqss 3547 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  C_  { Z } )
52 ssfi 7730 . . . 4  |-  ( ( { Z }  e.  Fin  /\  ran  ( F  |`  ( `' F " { Z } ) ) 
C_  { Z }
)  ->  ran  ( F  |`  ( `' F " { Z } ) )  e.  Fin )
5345, 51, 52sylancr 663 . . 3  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  e. 
Fin )
54 unfi 7776 . . 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 )
5544, 53, 54syl2anc 661 . 2  |-  ( ph  ->  ( ran  ( F  |`  ( `' F "
( _V  \  { Z } ) ) )  u.  ran  ( F  |`  ( `' F " { Z } ) ) )  e.  Fin )
5625, 55eqeltrd 2548 1  |-  ( ph  ->  ran  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    u. cun 3467    i^i cin 3468    C_ wss 3469   {csn 4020   class class class wbr 4440   `'ccnv 4991   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573    Fn wfn 5574   -onto->wfo 5577  (class class class)co 6275   supp csupp 6891    ~<_ cdom 7504   Fincfn 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-ac2 8832
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-fin 7510  df-card 8309  df-acn 8312  df-ac 8486
This theorem is referenced by:  fpwrelmapffslem  27077
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