Theorem lmff 14944

Description: If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)

Hypotheses

Ref	Expression
lmff.1	⊢ 𝑍 = (ℤ≥‘𝑀)
lmff.3	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
lmff.4	⊢ (𝜑 → 𝑀 ∈ ℤ)
lmff.5	⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))

Assertion

Ref	Expression
lmff	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)

Distinct variable groups:   𝑗,𝐹   𝑗,𝐽   𝑗,𝑀   𝜑,𝑗   𝑗,𝑋   𝑗,𝑍

Proof of Theorem lmff

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmff.5	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))
2		eldm2g 4922	. . . . . . 7 ⊢ (𝐹 ∈ dom (⇝𝑡‘𝐽) → (𝐹 ∈ dom (⇝𝑡‘𝐽) ↔ ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽)))
3	2	ibi 176	. . . . . 6 ⊢ (𝐹 ∈ dom (⇝𝑡‘𝐽) → ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
5		df-br 4084	. . . . . 6 ⊢ (𝐹(⇝𝑡‘𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
6	5	exbii 1651	. . . . 5 ⊢ (∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦 ↔ ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
7	4, 6	sylibr 134	. . . 4 ⊢ (𝜑 → ∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦)
8		lmff.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		lmcl 14940	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
10	8, 9	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
11		eleq2 2293	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (𝑦 ∈ 𝑗 ↔ 𝑦 ∈ 𝑋))
12		feq3 5461	. . . . . . . 8 ⊢ (𝑗 = 𝑋 → ((𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ (𝐹 ↾ 𝑥):𝑥⟶𝑋))
13	12	rexbidv 2531	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋))
14	11, 13	imbi12d 234	. . . . . 6 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗) ↔ (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)))
15	8	lmbr 14908	. . . . . . . 8 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑦 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗))))
16	15	biimpa 296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗)))
17	16	simp3d 1035	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗))
18		toponmax 14720	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
19	8, 18	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
20	19	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑋 ∈ 𝐽)
21	14, 17, 20	rspcdva 2912	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋))
22	10, 21	mpd 13	. . . 4 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
23	7, 22	exlimddv 1945	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
24		uzf 9741	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
25		ffn 5476	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
26		reseq2 5003	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (ℤ≥‘𝑗)))
27		id 19	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → 𝑥 = (ℤ≥‘𝑗))
28	26, 27	feq12d 5466	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → ((𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
29	28	rexrn 5777	. . . 4 ⊢ (ℤ≥ Fn ℤ → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
30	24, 25, 29	mp2b 8	. . 3 ⊢ (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)
31	23, 30	sylib 122	. 2 ⊢ (𝜑 → ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)
32		lmff.4	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33		lmff.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
34	33	rexuz3 11522	. . . 4 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
35	32, 34	syl 14	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
36	16	simp1d 1033	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm ℂ))
37	7, 36	exlimddv 1945	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
38		pmfun 6828	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 ↑pm ℂ) → Fun 𝐹)
39	37, 38	syl 14	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
40		ffvresb 5803	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
41	39, 40	syl 14	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
42	41	rexbidv 2531	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
43	41	rexbidv 2531	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
44	35, 42, 43	3bitr4d 220	. 2 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
45	31, 44	mpbird 167	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  𝒫 cpw 3649  ⟨cop 3669   class class class wbr 4083  dom cdm 4720  ran crn 4721   ↾ cres 4722  Fun wfun 5315   Fn wfn 5316  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010   ↑_pm cpm 6809  ℂcc 8013  ℤcz 9462  ℤ_≥cuz 9738  TopOnctopon 14705  ⇝_𝑡clm 14882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-pm 6811  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-top 14693  df-topon 14706  df-lm 14885

This theorem is referenced by: (None)