Theorem ffsrn 31941

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 5893	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		dfrn4 6198	. . . . . . 7 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
4	2, 3	eqtri 2760	. . . . . 6 ⊢ dom 𝐹 = (◡𝐹 “ V)
5		df-fn 6543	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)))
6		fnresdm 6666	. . . . . . 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 586	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (◡𝐹 “ V)) = 𝐹)
9		imaundi 6146	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))
10	9	reseq2i 5976	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍})))
11		undif1 4474	. . . . . . . . 9 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
12		ssv 4005	. . . . . . . . . 10 ⊢ {𝑍} ⊆ V
13		ssequn2 4182	. . . . . . . . . 10 ⊢ ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
14	12, 13	mpbi 229	. . . . . . . . 9 ⊢ (V ∪ {𝑍}) = V
15	11, 14	eqtri 2760	. . . . . . . 8 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = V
16	15	imaeq2i 6055	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (◡𝐹 “ V)
17	16	reseq2i 5976	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (◡𝐹 “ V))
18		resundi 5993	. . . . . 6 ⊢ (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
19	10, 17, 18	3eqtr3i 2768	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ V)) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
20	8, 19	eqtr3di 2787	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
21	20	rneqd 5935	. . 3 ⊢ (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
22		rnun 6142	. . 3 ⊢ ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍})))
23	21, 22	eqtrdi 2788	. 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 8155	. . . . . 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 2834	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30		cnvexg 7911	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
31		imaexg 7902	. . . . . 6 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
32	24, 30, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
33		cnvimass 6077	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34		fores 6812	. . . . . . 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 6804	. . . . . 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 584	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42		snfi 9040	. . . 4 ⊢ {𝑍} ∈ Fin
43		df-ima 5688	. . . . . 6 ⊢ (𝐹 “ (◡𝐹 “ {𝑍})) = ran (𝐹 ↾ (◡𝐹 “ {𝑍}))
44		funimacnv 6626	. . . . . . 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 2786	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47		inss1 4227	. . . . 5 ⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
48	46, 47	eqsstrdi 4035	. . . 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 587	. . 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 584	. 2 ⊢ (𝜑 → (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍}))) ∈ Fin)
53	23, 52	eqeltrd 2833	1 ⊢ (𝜑 → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  {csn 4627   class class class wbr 5147  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   ↾ cres 5677   “ cima 5678  Fun wfun 6534   Fn wfn 6535  –onto→wfo 6538  (class class class)co 7405   supp csupp 8142   ≼ cdom 8933  Fincfn 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-ac2 10454

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  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  31944