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

Step	Hyp	Ref	Expression
1		imaundi 5254	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) )
2	1	reseq2i 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 )
6	4, 5	mpbi 213	. . . . . . . . 9 |- ( _V u. { Z } ) = _V
7	3, 6	eqtri 2493	. . . . . . . 8 |- ( ( _V \ { Z } ) u. { Z } ) = _V
8	7	imaeq2i 5172	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( `' F " _V )
9	8	reseq2i 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 } ) ) )
11	2, 9, 10	3eqtr3i 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 )
15	13, 14	eqtri 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 )
18	16, 17	sylbir 218	. . . . . 6 |- ( ( Fun F /\ dom F = ( `' F " _V ) ) -> ( F |` ( `' F " _V ) ) = F )
19	12, 15, 18	sylancl 675	. . . . 5 |- ( ph -> ( F |` ( `' F " _V ) ) = F )
20	11, 19	syl5reqr 2520	. . . 4 |- ( ph -> F = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
21	20	rneqd 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 } ) ) )
23	21, 22	syl6eq 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 } ) ) )
27	24, 25, 26	syl2anc 673	. . . . 5 |- ( ph -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
28		ffsrn.2	. . . . 5 |- ( ph -> ( F supp Z ) e. Fin )
29	27, 28	eqeltrrd 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 )
32	24, 30, 31	3syl 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 } ) ) ) )
35	12, 33, 34	sylancl 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 } ) ) )
37	35, 36	syl 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 } ) ) ) )
39	32, 37, 38	sylc 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 )
41	29, 39, 40	syl2anc 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 ) )
45	12, 44	syl 17	. . . . . 6 |- ( ph -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
46	43, 45	syl5eqr 2519	. . . . 5 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
47		inss1 3643	. . . . 5 |- ( { Z } i^i ran F ) C_ { Z }
48	46, 47	syl6eqss 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 )
50	42, 48, 49	sylancr 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 )
52	41, 50, 51	syl2anc 673	. 2 |- ( ph -> ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) ) e. Fin )
53	23, 52	eqeltrd 2549	1 |- ( ph -> ran F e. Fin )

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