Theorem ffsrn 32741

Description: The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)

Hypotheses

Ref	Expression
ffsrn.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
ffsrn.0	⊢ (𝜑 → 𝐹 ∈ 𝑉)
ffsrn.1	⊢ (𝜑 → Fun 𝐹)
ffsrn.2	⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)

Assertion

Ref	Expression
ffsrn	⊢ (𝜑 → ran 𝐹 ∈ Fin)

Proof of Theorem ffsrn

Step	Hyp	Ref	Expression
1		ffsrn.1	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
2		dfdm4 5905	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		dfrn4 6221	. . . . . . 7 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
4	2, 3	eqtri 2764	. . . . . 6 ⊢ dom 𝐹 = (◡𝐹 “ V)
5		df-fn 6563	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)))
6		fnresdm 6686	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
7	5, 6	sylbir 235	. . . . . 6 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)) → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
8	1, 4, 7	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
9		imaundi 6168	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))
10	9	reseq2i 5993	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍})))
11		undif1 4475	. . . . . . . . 9 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
12		ssv 4007	. . . . . . . . . 10 ⊢ {𝑍} ⊆ V
13		ssequn2 4188	. . . . . . . . . 10 ⊢ ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
14	12, 13	mpbi 230	. . . . . . . . 9 ⊢ (V ∪ {𝑍}) = V
15	11, 14	eqtri 2764	. . . . . . . 8 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = V
16	15	imaeq2i 6075	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (◡𝐹 “ V)
17	16	reseq2i 5993	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (◡𝐹 “ V))
18		resundi 6010	. . . . . 6 ⊢ (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
19	10, 17, 18	3eqtr3i 2772	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ V)) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
20	8, 19	eqtr3di 2791	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
21	20	rneqd 5948	. . 3 ⊢ (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
22		rnun 6164	. . 3 ⊢ ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍})))
23	21, 22	eqtrdi 2792	. 2 ⊢ (𝜑 → ran 𝐹 = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))))
24		ffsrn.0	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
25		ffsrn.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
26		suppimacnv 8200	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
27	24, 25, 26	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
28		ffsrn.2	. . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
29	27, 28	eqeltrrd 2841	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30		cnvexg 7947	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
31		imaexg 7936	. . . . . 6 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
32	24, 30, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
33		cnvimass 6099	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34		fores 6829	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹) → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))):(◡𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (◡𝐹 “ (V ∖ {𝑍}))))
35	1, 33, 34	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))):(◡𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (◡𝐹 “ (V ∖ {𝑍}))))
36		fofn 6821	. . . . . 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 ⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) Fn (◡𝐹 “ (V ∖ {𝑍})))
38		fnrndomg 10577	. . . . 5 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) ∈ V → ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) Fn (◡𝐹 “ (V ∖ {𝑍})) → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍}))))
39	32, 37, 38	sylc 65	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍})))
40		domfi 9230	. . . 4 ⊢ (((◡𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
41	29, 39, 40	syl2anc 584	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42		snfi 9084	. . . 4 ⊢ {𝑍} ∈ Fin
43		df-ima 5697	. . . . . 6 ⊢ (𝐹 “ (◡𝐹 “ {𝑍})) = ran (𝐹 ↾ (◡𝐹 “ {𝑍}))
44		funimacnv 6646	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
45	1, 44	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
46	43, 45	eqtr3id 2790	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47		inss1 4236	. . . . 5 ⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
48	46, 47	eqsstrdi 4027	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍})
49		ssfi 9214	. . . 4 ⊢ (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin)
50	42, 48, 49	sylancr 587	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin)
51		unfi 9212	. . 3 ⊢ ((ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin) → (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) ∈ Fin)
52	41, 50, 51	syl2anc 584	. 2 ⊢ (𝜑 → (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) ∈ Fin)
53	23, 52	eqeltrd 2840	1 ⊢ (𝜑 → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ∖ cdif 3947   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  {csn 4625   class class class wbr 5142  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   ↾ cres 5686   “ cima 5687  Fun wfun 6554   Fn wfn 6555  –onto→wfo 6558  (class class class)co 7432   supp csupp 8186   ≼ cdom 8984  Fincfn 8986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-ac2 10504

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-fin 8990  df-card 9980  df-acn 9983  df-ac 10157

This theorem is referenced by:  fpwrelmapffslem  32744