Theorem ffsrn 29504

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 5545	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) )
2	1	reseq2i 5393	. . . . . 6 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( F |` ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) ) )
3		undif1 4043	. . . . . . . . 9 |- ( ( _V \ { Z } ) u. { Z } ) = ( _V u. { Z } )
4		ssv 3625	. . . . . . . . . 10 |- { Z } C_ _V
5		ssequn2 3786	. . . . . . . . . 10 |- ( { Z } C_ _V <-> ( _V u. { Z } ) = _V )
6	4, 5	mpbi 220	. . . . . . . . 9 |- ( _V u. { Z } ) = _V
7	3, 6	eqtri 2644	. . . . . . . 8 |- ( ( _V \ { Z } ) u. { Z } ) = _V
8	7	imaeq2i 5464	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( `' F " _V )
9	8	reseq2i 5393	. . . . . 6 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( F |` ( `' F " _V ) )
10		resundi 5410	. . . . . 6 |- ( F |` ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
11	2, 9, 10	3eqtr3i 2652	. . . . 5 |- ( F |` ( `' F " _V ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
12		ffsrn.1	. . . . . 6 |- ( ph -> Fun F )
13		dfdm4 5316	. . . . . . 7 |- dom F = ran `' F
14		dfrn4 5595	. . . . . . 7 |- ran `' F = ( `' F " _V )
15	13, 14	eqtri 2644	. . . . . 6 |- dom F = ( `' F " _V )
16		df-fn 5891	. . . . . . 7 |- ( F Fn ( `' F " _V ) <-> ( Fun F /\ dom F = ( `' F " _V ) ) )
17		fnresdm 6000	. . . . . . 7 |- ( F Fn ( `' F " _V ) -> ( F |` ( `' F " _V ) ) = F )
18	16, 17	sylbir 225	. . . . . 6 |- ( ( Fun F /\ dom F = ( `' F " _V ) ) -> ( F |` ( `' F " _V ) ) = F )
19	12, 15, 18	sylancl 694	. . . . 5 |- ( ph -> ( F |` ( `' F " _V ) ) = F )
20	11, 19	syl5reqr 2671	. . . 4 |- ( ph -> F = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
21	20	rneqd 5353	. . 3 |- ( ph -> ran F = ran ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
22		rnun 5541	. . 3 |- ran ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) = ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) )
23	21, 22	syl6eq 2672	. 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 7306	. . . . . 6 |- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
27	24, 25, 26	syl2anc 693	. . . . 5 |- ( ph -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
28		ffsrn.2	. . . . 5 |- ( ph -> ( F supp Z ) e. Fin )
29	27, 28	eqeltrrd 2702	. . . 4 |- ( ph -> ( `' F " ( _V \ { Z } ) ) e. Fin )
30		cnvexg 7112	. . . . . 6 |- ( F e. V -> `' F e. _V )
31		imaexg 7103	. . . . . 6 |- ( `' F e. _V -> ( `' F " ( _V \ { Z } ) ) e. _V )
32	24, 30, 31	3syl 18	. . . . 5 |- ( ph -> ( `' F " ( _V \ { Z } ) ) e. _V )
33		cnvimass 5485	. . . . . . 7 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
34		fores 6124	. . . . . . 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 694	. . . . . 6 |- ( ph -> ( F |` ( `' F " ( _V \ { Z } ) ) ) : ( `' F " ( _V \ { Z } ) ) -onto-> ( F " ( `' F " ( _V \ { Z } ) ) ) )
36		fofn 6117	. . . . . 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 9358	. . . . 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 65	. . . 4 |- ( ph -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) ~<_ ( `' F " ( _V \ { Z } ) ) )
40		domfi 8181	. . . 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 693	. . 3 |- ( ph -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) e. Fin )
42		snfi 8038	. . . 4 |- { Z } e. Fin
43		df-ima 5127	. . . . . 6 |- ( F " ( `' F " { Z } ) ) = ran ( F |` ( `' F " { Z } ) )
44		funimacnv 5970	. . . . . . 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 2670	. . . . 5 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
47		inss1 3833	. . . . 5 |- ( { Z } i^i ran F ) C_ { Z }
48	46, 47	syl6eqss 3655	. . . 4 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) C_ { Z } )
49		ssfi 8180	. . . 4 |- ( ( { Z } e. Fin /\ ran ( F |` ( `' F " { Z } ) ) C_ { Z } ) -> ran ( F |` ( `' F " { Z } ) ) e. Fin )
50	42, 48, 49	sylancr 695	. . 3 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) e. Fin )
51		unfi 8227	. . 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 693	. 2 |- ( ph -> ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) ) e. Fin )
53	23, 52	eqeltrd 2701	1 |- ( ph -> ran F e. Fin )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886  (class class class)co 6650   supp csupp 7295   ~<_ cdom 7953   Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by:  fpwrelmapffslem  29507