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Theorem ffsrn 26173
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 3855 . . . . . . . . 9  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  ( _V  u.  { Z } )
2 ssv 3477 . . . . . . . . . 10  |-  { Z }  C_  _V
3 ssequn2 3630 . . . . . . . . . 10  |-  ( { Z }  C_  _V  <->  ( _V  u.  { Z } )  =  _V )
42, 3mpbi 208 . . . . . . . . 9  |-  ( _V  u.  { Z }
)  =  _V
51, 4eqtri 2480 . . . . . . . 8  |-  ( ( _V  \  { Z } )  u.  { Z } )  =  _V
65imaeq2i 5268 . . . . . . 7  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( `' F " _V )
76reseq2i 5208 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( `' F " _V ) )
8 imaundi 5350 . . . . . . . 8  |-  ( `' F " ( ( _V  \  { Z } )  u.  { Z } ) )  =  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) )
98reseq2i 5208 . . . . . . 7  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( F  |`  ( ( `' F " ( _V 
\  { Z }
) )  u.  ( `' F " { Z } ) ) )
10 resundi 5225 . . . . . . 7  |-  ( F  |`  ( ( `' F " ( _V  \  { Z } ) )  u.  ( `' F " { Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
119, 10eqtri 2480 . . . . . 6  |-  ( F  |`  ( `' F "
( ( _V  \  { Z } )  u. 
{ Z } ) ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
127, 11eqtr3i 2482 . . . . 5  |-  ( F  |`  ( `' F " _V ) )  =  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) )
13 ffsrn.1 . . . . . 6  |-  ( ph  ->  Fun  F )
14 dfdm4 5133 . . . . . . 7  |-  dom  F  =  ran  `' F
15 dfrn4 5399 . . . . . . 7  |-  ran  `' F  =  ( `' F " _V )
1614, 15eqtri 2480 . . . . . 6  |-  dom  F  =  ( `' F " _V )
17 df-fn 5522 . . . . . . 7  |-  ( F  Fn  ( `' F " _V )  <->  ( Fun  F  /\  dom  F  =  ( `' F " _V ) ) )
18 fnresdm 5621 . . . . . . 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 2507 . . . 4  |-  ( ph  ->  F  =  ( ( F  |`  ( `' F " ( _V  \  { Z } ) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
2221rneqd 5168 . . 3  |-  ( ph  ->  ran  F  =  ran  ( ( F  |`  ( `' F " ( _V 
\  { Z }
) ) )  u.  ( F  |`  ( `' F " { Z } ) ) ) )
23 rnun 5346 . . . 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 2492 . 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 6804 . . . . . 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 2540 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
Fin )
32 cnvimass 5290 . . . . . . 7  |-  ( `' F " ( _V 
\  { Z }
) )  C_  dom  F
33 fores 5730 . . . . . . 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 5723 . . . . . 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 6627 . . . . . . 7  |-  ( F  e.  V  ->  `' F  e.  _V )
38 imaexg 6618 . . . . . . 7  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  { Z }
) )  e.  _V )
3926, 37, 383syl 20 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  { Z } ) )  e. 
_V )
40 fnrndomg 8806 . . . . . 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 7638 . . . 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 7493 . . . 4  |-  { Z }  e.  Fin
46 df-ima 4954 . . . . . 6  |-  ( F
" ( `' F " { Z } ) )  =  ran  ( F  |`  ( `' F " { Z } ) )
47 funimacnv 5591 . . . . . . 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 2506 . . . . 5  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  =  ( { Z }  i^i  ran  F ) )
50 inss1 3671 . . . . 5  |-  ( { Z }  i^i  ran  F )  C_  { Z }
5149, 50syl6eqss 3507 . . . 4  |-  ( ph  ->  ran  ( F  |`  ( `' F " { Z } ) )  C_  { Z } )
52 ssfi 7637 . . . 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 7683 . . 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 2539 1  |-  ( ph  ->  ran  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071    \ cdif 3426    u. cun 3427    i^i cin 3428    C_ wss 3429   {csn 3978   class class class wbr 4393   `'ccnv 4940   dom cdm 4941   ran crn 4942    |` cres 4943   "cima 4944   Fun wfun 5513    Fn wfn 5514   -onto->wfo 5517  (class class class)co 6193   supp csupp 6793    ~<_ cdom 7411   Fincfn 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-ac2 8736
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-fin 7417  df-card 8213  df-acn 8216  df-ac 8390
This theorem is referenced by:  fpwrelmapffslem  26176
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