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

Step	Hyp	Ref	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 )
4	2, 3	mpbi 208	. . . . . . . . 9 |- ( _V u. { Z } ) = _V
5	1, 4	eqtri 2480	. . . . . . . 8 |- ( ( _V \ { Z } ) u. { Z } ) = _V
6	5	imaeq2i 5268	. . . . . . 7 |- ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) = ( `' F " _V )
7	6	reseq2i 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 } ) )
9	8	reseq2i 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 } ) ) )
11	9, 10	eqtri 2480	. . . . . 6 |- ( F |` ( `' F " ( ( _V \ { Z } ) u. { Z } ) ) ) = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) )
12	7, 11	eqtr3i 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 )
16	14, 15	eqtri 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 )
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 2507	. . . 4 |- ( ph -> F = ( ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ( F |` ( `' F " { Z } ) ) ) )
22	21	rneqd 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 } ) ) )
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 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 } ) ) )
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 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 } ) ) ) )
34	13, 32, 33	sylancl 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 } ) ) )
36	34, 35	syl 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 )
39	26, 37, 38	3syl 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 } ) ) ) )
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 7638	. . . 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 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 ) )
48	13, 47	syl 16	. . . . . 6 |- ( ph -> ( F " ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
49	46, 48	syl5eqr 2506	. . . . 5 |- ( ph -> ran ( F |` ( `' F " { Z } ) ) = ( { Z } i^i ran F ) )
50		inss1 3671	. . . . 5 |- ( { Z } i^i ran F ) C_ { Z }
51	49, 50	syl6eqss 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 )
53	45, 51, 52	sylancr 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 )
55	44, 53, 54	syl2anc 661	. 2 |- ( ph -> ( ran ( F |` ( `' F " ( _V \ { Z } ) ) ) u. ran ( F |` ( `' F " { Z } ) ) ) e. Fin )
56	25, 55	eqeltrd 2539	1 |- ( ph -> ran F e. Fin )

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