Theorem ffsrn 32423

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 5885	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		dfrn4 6191	. . . . . . 7 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
4	2, 3	eqtri 2752	. . . . . 6 ⊢ dom 𝐹 = (◡𝐹 “ V)
5		df-fn 6536	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)))
6		fnresdm 6659	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
7	5, 6	sylbir 234	. . . . . 6 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)) → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
8	1, 4, 7	sylancl 585	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
9		imaundi 6139	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))
10	9	reseq2i 5968	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍})))
11		undif1 4467	. . . . . . . . 9 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
12		ssv 3998	. . . . . . . . . 10 ⊢ {𝑍} ⊆ V
13		ssequn2 4175	. . . . . . . . . 10 ⊢ ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
14	12, 13	mpbi 229	. . . . . . . . 9 ⊢ (V ∪ {𝑍}) = V
15	11, 14	eqtri 2752	. . . . . . . 8 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = V
16	15	imaeq2i 6047	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (◡𝐹 “ V)
17	16	reseq2i 5968	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (◡𝐹 “ V))
18		resundi 5985	. . . . . 6 ⊢ (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
19	10, 17, 18	3eqtr3i 2760	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ V)) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
20	8, 19	eqtr3di 2779	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
21	20	rneqd 5927	. . 3 ⊢ (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
22		rnun 6135	. . 3 ⊢ ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍})))
23	21, 22	eqtrdi 2780	. 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 8153	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
27	24, 25, 26	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
28		ffsrn.2	. . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
29	27, 28	eqeltrrd 2826	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30		cnvexg 7908	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
31		imaexg 7899	. . . . . 6 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
32	24, 30, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
33		cnvimass 6070	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34		fores 6805	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))):(◡𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (◡𝐹 “ (V ∖ {𝑍}))))
36		fofn 6797	. . . . . 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 10527	. . . . 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 9188	. . . 4 ⊢ (((◡𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ≼ (◡𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
41	29, 39, 40	syl2anc 583	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42		snfi 9040	. . . 4 ⊢ {𝑍} ∈ Fin
43		df-ima 5679	. . . . . 6 ⊢ (𝐹 “ (◡𝐹 “ {𝑍})) = ran (𝐹 ↾ (◡𝐹 “ {𝑍}))
44		funimacnv 6619	. . . . . . 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 2778	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47		inss1 4220	. . . . 5 ⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
48	46, 47	eqsstrdi 4028	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍})
49		ssfi 9169	. . . 4 ⊢ (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin)
50	42, 48, 49	sylancr 586	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ∈ Fin)
51		unfi 9168	. . 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 583	. 2 ⊢ (𝜑 → (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) ∈ Fin)
53	23, 52	eqeltrd 2825	1 ⊢ (𝜑 → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∖ cdif 3937   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  {csn 4620   class class class wbr 5138  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669  Fun wfun 6527   Fn wfn 6528  –onto→wfo 6531  (class class class)co 7401   supp csupp 8140   ≼ cdom 8933  Fincfn 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-ac2 10454

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-fin 8939  df-card 9930  df-acn 9933  df-ac 10107

This theorem is referenced by:  fpwrelmapffslem  32426