Theorem ffsrn 25980

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		undif1 3749	. . . . . . . . 9 |- ( ( _V \ { Z } ) u. { Z } ) = ( _V u. { Z } )
2		ssv 3371	. . . . . . . . . 10 |- { Z } C_ _V
3		ssequn2 3524	. . . . . . . . . 10 |- ( { Z } C_ _V <-> ( _V u. { Z } ) = _V )
4	2, 3	mpbi 208	. . . . . . . . 9 |- ( _V u. { Z } ) = _V
5	1, 4	eqtri 2458	. . . . . . . 8 |- ( ( _V \ { Z } ) u. { Z } ) = _V
6	5	imaeq2i 5162	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( `' F " _V )
7	6	reseq2i 5102	. . . . . 6 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( F |` ( `' F " _V ) )
8		imaundi 5244	. . . . . . . 8 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) )
9	8	reseq2i 5102	. . . . . . 7 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( F |` ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) ) )
10		resundi 5119	. . . . . . 7 |- ( F |` ( ( `' F " ( _V \ { Z } ) ) u. ( `' F " { Z } ) ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
11	9, 10	eqtri 2458	. . . . . 6 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
12	7, 11	eqtr3i 2460	. . . . 5 |- ( F |` ( `' F " _V ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
13		ffsrn.1	. . . . . 6 |- ( ph -> Fun F )
14		dfdm4 5027	. . . . . . 7 |- dom F = ran `' F
15		dfrn4 5293	. . . . . . 7 |- ran `' F = ( `' F " _V )
16	14, 15	eqtri 2458	. . . . . 6 |- dom F = ( `' F " _V )
17		df-fn 5416	. . . . . . 7 |- ( F Fn ( `' F " _V ) <-> ( Fun F /\ dom F = ( `' F " _V ) ) )
18		fnresdm 5515	. . . . . . 7 |- ( F Fn ( `' F " _V ) -> ( F |` ( `' F " _V ) ) = F )
19	17, 18	sylbir 213	. . . . . 6 |- ( ( Fun F /\ dom F = ( `' F " _V ) ) -> ( F |` ( `' F " _V ) ) = F )
20	13, 16, 19	sylancl 662	. . . . 5 |- ( ph -> ( F |` ( `' F " _V ) ) = F )
21	12, 20	syl5reqr 2485	. . . 4 |- ( ph -> F = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
22	21	rneqd 5062	. . 3 |- ( ph -> ran F = ran ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
23		rnun 5240	. . . 4 |- ran ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) = ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) )
24	23	a1i 11	. . 3 |- ( ph -> ran ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) = ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) ) )
25	22, 24	eqtrd 2470	. 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 6696	. . . . . 6 |- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
29	26, 27, 28	syl2anc 661	. . . . 5 |- ( ph -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
30		ffsrn.2	. . . . 5 |- ( ph -> ( F supp Z ) e. Fin )
31	29, 30	eqeltrrd 2513	. . . 4 |- ( ph -> ( `' F " ( _V \ { Z } ) ) e. Fin )
32		cnvimass 5184	. . . . . . 7 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
33		fores 5624	. . . . . . 7 |- ( ( Fun F /\ ( `' F " ( _V \ { Z } ) ) C_ dom F ) -> ( F |` ( `' F " ( _V \ { Z } ) ) ) : ( `' F " ( _V \ { Z } ) ) -onto-> ( F " ( `' F " ( _V \ { Z } ) ) ) )
34	13, 32, 33	sylancl 662	. . . . . 6 |- ( ph -> ( F |` ( `' F " ( _V \ { Z } ) ) ) : ( `' F " ( _V \ { Z } ) ) -onto-> ( F " ( `' F " ( _V \ { Z } ) ) ) )
35		fofn 5617	. . . . . 6 |- ( ( F |` ( `' F " ( _V \ { Z } ) ) ) : ( `' F " ( _V \ { Z } ) ) -onto-> ( F " ( `' F " ( _V \ { Z } ) ) ) -> ( F |` ( `' F " ( _V \ { Z } ) ) ) Fn ( `' F " ( _V \ { Z } ) ) )
36	34, 35	syl 16	. . . . 5 |- ( ph -> ( F |` ( `' F " ( _V \ { Z } ) ) ) Fn ( `' F " ( _V \ { Z } ) ) )
37		cnvexg 6519	. . . . . . 7 |- ( F e. V -> `' F e. _V )
38		imaexg 6510	. . . . . . 7 |- ( `' F e. _V -> ( `' F " ( _V \ { Z } ) ) e. _V )
39	26, 37, 38	3syl 20	. . . . . 6 |- ( ph -> ( `' F " ( _V \ { Z } ) ) e. _V )
40		fnrndomg 8694	. . . . . 6 |- ( ( `' F " ( _V \ { Z } ) ) e. _V -> ( ( F |` ( `' F " ( _V \ { Z } ) ) ) Fn ( `' F " ( _V \ { Z } ) ) -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) ~<_ ( `' F " ( _V \ { Z } ) ) ) )
41	39, 40	syl 16	. . . . 5 |- ( ph -> ( ( F |` ( `' F " ( _V \ { Z } ) ) ) Fn ( `' F " ( _V \ { Z } ) ) -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) ~<_ ( `' F " ( _V \ { Z } ) ) ) )
42	36, 41	mpd 15	. . . 4 |- ( ph -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) ~<_ ( `' F " ( _V \ { Z } ) ) )
43		domfi 7526	. . . 4 |- ( ( ( `' F " ( _V \ { Z } ) ) e. Fin /\ ran ( F |` ( `' F " ( _V \ { Z } ) ) ) ~<_ ( `' F " ( _V \ { Z } ) ) ) -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) e. Fin )
44	31, 42, 43	syl2anc 661	. . 3 |- ( ph -> ran ( F |` ( `' F " ( _V \ { Z } ) ) ) e. Fin )
45		snfi 7382	. . . 4 |- { Z } e. Fin
46		df-ima 4848	. . . . . 6 |- ( F " ( `' F " { Z } ) ) = ran ( F |` ( `' F " { Z } ) )
47		funimacnv 5485	. . . . . . 7 |- ( Fun F -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
48	13, 47	syl 16	. . . . . 6 |- ( ph -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
49	46, 48	syl5eqr 2484	. . . . 5 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
50		inss1 3565	. . . . 5 |- ( { Z } i^i ran F ) C_ { Z }
51	49, 50	syl6eqss 3401	. . . 4 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) C_ { Z } )
52		ssfi 7525	. . . 4 |- ( ( { Z } e. Fin /\ ran ( F |` ( `' F " { Z } ) ) C_ { Z } ) -> ran ( F |` ( `' F " { Z } ) ) e. Fin )
53	45, 51, 52	sylancr 663	. . 3 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) e. Fin )
54		unfi 7571	. . 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 )
55	44, 53, 54	syl2anc 661	. 2 |- ( ph -> ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) ) e. Fin )
56	25, 55	eqeltrd 2512	1 |- ( ph -> ran F e. Fin )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2967   \ cdif 3320   u. cun 3321   i^i cin 3322   C_ wss 3323   {csn 3872   class class class wbr 4287   `'ccnv 4834   dom cdm 4835   ran crn 4836   |` cres 4837   "cima 4838   Fun wfun 5407   Fn wfn 5408   -onto->wfo 5411  (class class class)co 6086   supp csupp 6685   ~<_ cdom 7300   Fincfn 7302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-ac2 8624

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-fin 7306  df-card 8101  df-acn 8104  df-ac 8278

This theorem is referenced by:  fpwrelmapffslem  25983