Theorem lmff 23325

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 5913	. . . . . . 7 ⊢ (𝐹 ∈ dom (⇝𝑡‘𝐽) → (𝐹 ∈ dom (⇝𝑡‘𝐽) ↔ ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽)))
3	2	ibi 267	. . . . . 6 ⊢ (𝐹 ∈ dom (⇝𝑡‘𝐽) → ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
5		df-br 5149	. . . . . 6 ⊢ (𝐹(⇝𝑡‘𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
6	5	exbii 1845	. . . . 5 ⊢ (∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦 ↔ ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
7	4, 6	sylibr 234	. . . 4 ⊢ (𝜑 → ∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦)
8		lmff.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		lmcl 23321	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
10	8, 9	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
11		eleq2 2828	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (𝑦 ∈ 𝑗 ↔ 𝑦 ∈ 𝑋))
12		feq3 6719	. . . . . . . 8 ⊢ (𝑗 = 𝑋 → ((𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ (𝐹 ↾ 𝑥):𝑥⟶𝑋))
13	12	rexbidv 3177	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋))
14	11, 13	imbi12d 344	. . . . . 6 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗) ↔ (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)))
15	8	lmbr 23282	. . . . . . . 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 476	. . . . . . 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 1143	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗))
18		toponmax 22948	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
19	8, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑋 ∈ 𝐽)
21	14, 17, 20	rspcdva 3623	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋))
22	10, 21	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
23	7, 22	exlimddv 1933	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
24		uzf 12879	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
25		ffn 6737	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
26		reseq2 5995	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (ℤ≥‘𝑗)))
27		id 22	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → 𝑥 = (ℤ≥‘𝑗))
28	26, 27	feq12d 6725	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → ((𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
29	28	rexrn 7107	. . . 4 ⊢ (ℤ≥ Fn ℤ → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
30	24, 25, 29	mp2b 10	. . 3 ⊢ (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)
31	23, 30	sylib 218	. 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 15384	. . . 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 17	. . 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 1141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm ℂ))
37	7, 36	exlimddv 1933	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
38		pmfun 8886	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 ↑pm ℂ) → Fun 𝐹)
39	37, 38	syl 17	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
40		ffvresb 7145	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
41	39, 40	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
42	41	rexbidv 3177	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
43	41	rexbidv 3177	. . 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 311	. 2 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
45	31, 44	mpbird 257	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  𝒫 cpw 4605  ⟨cop 4637   class class class wbr 5148  dom cdm 5689  ran crn 5690   ↾ cres 5691  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_pm cpm 8866  ℂcc 11151  ℤcz 12611  ℤ_≥cuz 12876  TopOnctopon 22932  ⇝_𝑡clm 23250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-neg 11493  df-z 12612  df-uz 12877  df-top 22916  df-topon 22933  df-lm 23253

This theorem is referenced by:  lmle  25349