Theorem lmff 22452

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 5808	. . . . . . 7 ⊢ (𝐹 ∈ dom (⇝𝑡‘𝐽) → (𝐹 ∈ dom (⇝𝑡‘𝐽) ↔ ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽)))
3	2	ibi 266	. . . . . 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 5075	. . . . . 6 ⊢ (𝐹(⇝𝑡‘𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
6	5	exbii 1850	. . . . 5 ⊢ (∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦 ↔ ∃𝑦⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
7	4, 6	sylibr 233	. . . 4 ⊢ (𝜑 → ∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦)
8		lmff.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		lmcl 22448	. . . . . 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 2827	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (𝑦 ∈ 𝑗 ↔ 𝑦 ∈ 𝑋))
12		feq3 6583	. . . . . . . 8 ⊢ (𝑗 = 𝑋 → ((𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ (𝐹 ↾ 𝑥):𝑥⟶𝑋))
13	12	rexbidv 3226	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋))
14	11, 13	imbi12d 345	. . . . . 6 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗) ↔ (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)))
15	8	lmbr 22409	. . . . . . . 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 477	. . . . . . 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 22075	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
19	8, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
20	19	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑋 ∈ 𝐽)
21	14, 17, 20	rspcdva 3562	. . . . 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 1938	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
24		uzf 12585	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
25		ffn 6600	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
26		reseq2 5886	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (ℤ≥‘𝑗)))
27		id 22	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → 𝑥 = (ℤ≥‘𝑗))
28	26, 27	feq12d 6588	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → ((𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
29	28	rexrn 6963	. . . 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 217	. 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 15060	. . . 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 1938	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
38		pmfun 8635	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 ↑pm ℂ) → Fun 𝐹)
39	37, 38	syl 17	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
40		ffvresb 6998	. . . . 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 3226	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
43	41	rexbidv 3226	. . 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 256	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  𝒫 cpw 4533  ⟨cop 4567   class class class wbr 5074  dom cdm 5589  ran crn 5590   ↾ cres 5591  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_pm cpm 8616  ℂcc 10869  ℤcz 12319  ℤ_≥cuz 12582  TopOnctopon 22059  ⇝_𝑡clm 22377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-neg 11208  df-z 12320  df-uz 12583  df-top 22043  df-topon 22060  df-lm 22380

This theorem is referenced by:  lmle  24465