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

Step	Hyp	Ref	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 )
4	2, 3	mpbi 208	. . . . . . . . 9 |- ( _V u. { Z } ) = _V
5	1, 4	eqtri 2489	. . . . . . . 8 |- ( ( _V \ { Z } ) u. { Z } ) = _V
6	5	imaeq2i 5326	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( `' F " _V )
7	6	reseq2i 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 } ) )
9	8	reseq2i 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 } ) ) )
11	9, 10	eqtri 2489	. . . . . 6 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
12	7, 11	eqtr3i 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 )
16	14, 15	eqtri 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 )
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 2516	. . . 4 |- ( ph -> F = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
22	21	rneqd 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 } ) ) )
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 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 } ) ) )
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 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 } ) ) ) )
34	13, 32, 33	sylancl 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 } ) ) )
36	34, 35	syl 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 )
39	26, 37, 38	3syl 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 } ) ) ) )
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 7731	. . . 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 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 ) )
48	13, 47	syl 16	. . . . . 6 |- ( ph -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
49	46, 48	syl5eqr 2515	. . . . 5 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
50		inss1 3711	. . . . 5 |- ( { Z } i^i ran F ) C_ { Z }
51	49, 50	syl6eqss 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 )
53	45, 51, 52	sylancr 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 )
55	44, 53, 54	syl2anc 661	. 2 |- ( ph -> ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) ) e. Fin )
56	25, 55	eqeltrd 2548	1 |- ( ph -> ran F e. Fin )

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