Theorem lmff 23257

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 5856	. . . . . . 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 5101	. . . . . 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 234	. . . 4 ⊢ (𝜑 → ∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦)
8		lmff.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		lmcl 23253	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
10	8, 9	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
11		eleq2 2826	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (𝑦 ∈ 𝑗 ↔ 𝑦 ∈ 𝑋))
12		feq3 6650	. . . . . . . 8 ⊢ (𝑗 = 𝑋 → ((𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ (𝐹 ↾ 𝑥):𝑥⟶𝑋))
13	12	rexbidv 3162	. . . . . . 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 23214	. . . . . . . 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 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗))
18		toponmax 22882	. . . . . . . 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 3579	. . . . 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 1937	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
24		uzf 12766	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
25		ffn 6670	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
26		reseq2 5941	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (ℤ≥‘𝑗)))
27		id 22	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → 𝑥 = (ℤ≥‘𝑗))
28	26, 27	feq12d 6658	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → ((𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
29	28	rexrn 7041	. . . 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 15284	. . . 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 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm ℂ))
37	7, 36	exlimddv 1937	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
38		pmfun 8796	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 ↑pm ℂ) → Fun 𝐹)
39	37, 38	syl 17	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
40		ffvresb 7080	. . . . 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 3162	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
43	41	rexbidv 3162	. . 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  𝒫 cpw 4556  ⟨cop 4588   class class class wbr 5100  dom cdm 5632  ran crn 5633   ↾ cres 5634  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_pm cpm 8776  ℂcc 11036  ℤcz 12500  ℤ_≥cuz 12763  TopOnctopon 22866  ⇝_𝑡clm 23182

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-neg 11379  df-z 12501  df-uz 12764  df-top 22850  df-topon 22867  df-lm 23185

This theorem is referenced by:  lmle  25269