Theorem ffsrn 32715

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 5839	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		dfrn4 6154	. . . . . . 7 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
4	2, 3	eqtri 2756	. . . . . 6 ⊢ dom 𝐹 = (◡𝐹 “ V)
5		df-fn 6489	. . . . . . 7 ⊢ (𝐹 Fn (◡𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (◡𝐹 “ V)))
6		fnresdm 6605	. . . . . . 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 6101	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))
10	9	reseq2i 5929	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍})))
11		undif1 4425	. . . . . . . . 9 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
12		ssv 3955	. . . . . . . . . 10 ⊢ {𝑍} ⊆ V
13		ssequn2 4138	. . . . . . . . . 10 ⊢ ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
14	12, 13	mpbi 230	. . . . . . . . 9 ⊢ (V ∪ {𝑍}) = V
15	11, 14	eqtri 2756	. . . . . . . 8 ⊢ ((V ∖ {𝑍}) ∪ {𝑍}) = V
16	15	imaeq2i 6011	. . . . . . 7 ⊢ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (◡𝐹 “ V)
17	16	reseq2i 5929	. . . . . 6 ⊢ (𝐹 ↾ (◡𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (◡𝐹 “ V))
18		resundi 5946	. . . . . 6 ⊢ (𝐹 ↾ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐹 “ {𝑍}))) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
19	10, 17, 18	3eqtr3i 2764	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ V)) = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍})))
20	8, 19	eqtr3di 2783	. . . 4 ⊢ (𝜑 → 𝐹 = ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
21	20	rneqd 5882	. . 3 ⊢ (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))))
22		rnun 6097	. . 3 ⊢ ran ((𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (◡𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (◡𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (◡𝐹 “ {𝑍})))
23	21, 22	eqtrdi 2784	. 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 8110	. . . . . 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 7860	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
31		imaexg 7849	. . . . . 6 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
32	24, 30, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ V)
33		cnvimass 6035	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34		fores 6750	. . . . . . 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 6742	. . . . . 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 10434	. . . . 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 9105	. . . 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 8972	. . . 4 ⊢ {𝑍} ∈ Fin
43		df-ima 5632	. . . . . 6 ⊢ (𝐹 “ (◡𝐹 “ {𝑍})) = ran (𝐹 ↾ (◡𝐹 “ {𝑍}))
44		funimacnv 6567	. . . . . . 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 2782	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47		inss1 4186	. . . . 5 ⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
48	46, 47	eqsstrdi 3975	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (◡𝐹 “ {𝑍})) ⊆ {𝑍})
49		ssfi 9089	. . . 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 9087	. . 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 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  {csn 4575   class class class wbr 5093  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6480   Fn wfn 6481  –onto→wfo 6484  (class class class)co 7352   supp csupp 8096   ≼ cdom 8873  Fincfn 8875

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-ac2 10361

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-1o 8391  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-fin 8879  df-card 9839  df-acn 9842  df-ac 10014

This theorem is referenced by:  fpwrelmapffslem  32719