Theorem ffsrn 32709

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 5835	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		dfrn4 6149	. . . . . . 7 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
4	2, 3	eqtri 2754	. . . . . 6 ⊢ dom 𝐹 = (◡𝐹 “ V)
5		df-fn 6484	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)))
6		fnresdm 6600	. . . . . . 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 6096	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))
10	9	reseq2i 5925	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍})))
11		undif1 4426	. . . . . . . . 9 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
12		ssv 3959	. . . . . . . . . 10 ⊢ {𝑍} ⊆ V
13		ssequn2 4139	. . . . . . . . . 10 ⊢ ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
14	12, 13	mpbi 230	. . . . . . . . 9 ⊢ (V ∪ {𝑍}) = V
15	11, 14	eqtri 2754	. . . . . . . 8 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = V
16	15	imaeq2i 6007	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (◡𝐹 “ V)
17	16	reseq2i 5925	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (◡𝐹 “ V))
18		resundi 5942	. . . . . 6 ⊢ (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
19	10, 17, 18	3eqtr3i 2762	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ V)) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
20	8, 19	eqtr3di 2781	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
21	20	rneqd 5878	. . 3 ⊢ (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
22		rnun 6092	. . 3 ⊢ ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍})))
23	21, 22	eqtrdi 2782	. 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 8104	. . . . . 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 2832	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30		cnvexg 7854	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
31		imaexg 7843	. . . . . 6 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
32	24, 30, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
33		cnvimass 6031	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34		fores 6745	. . . . . . 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 6737	. . . . . 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 10427	. . . . 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 9098	. . . 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 8965	. . . 4 ⊢ {𝑍} ∈ Fin
43		df-ima 5629	. . . . . 6 ⊢ (𝐹 “ (◡𝐹 “ {𝑍})) = ran (𝐹 ↾ (◡𝐹 “ {𝑍}))
44		funimacnv 6562	. . . . . . 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 2780	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47		inss1 4187	. . . . 5 ⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
48	46, 47	eqsstrdi 3979	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍})
49		ssfi 9082	. . . 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 9080	. . 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 2831	1 ⊢ (𝜑 → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  {csn 4576   class class class wbr 5091  ^◡ccnv 5615  dom cdm 5616  ran crn 5617   ↾ cres 5618   “ cima 5619  Fun wfun 6475   Fn wfn 6476  –onto→wfo 6479  (class class class)co 7346   supp csupp 8090   ≼ cdom 8867  Fincfn 8869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-ac2 10354

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-fin 8873  df-card 9832  df-acn 9835  df-ac 10007

This theorem is referenced by:  fpwrelmapffslem  32713