Theorem lmff 14765

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 4879	. . . . . . 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 4048	. . . . . 6 ⊢ (𝐹(⇝𝑡‘𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡‘𝐽))
6	5	exbii 1629	. . . . 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 14761	. . . . . 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 2270	. . . . . . 7 ⊢ (𝑗 = 𝑋 → (𝑦 ∈ 𝑗 ↔ 𝑦 ∈ 𝑋))
12		feq3 5416	. . . . . . . 8 ⊢ (𝑗 = 𝑋 → ((𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ (𝐹 ↾ 𝑥):𝑥⟶𝑋))
13	12	rexbidv 2508	. . . . . . 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 14729	. . . . . . . 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 1014	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗))
18		toponmax 14541	. . . . . . . 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 2883	. . . . 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 1923	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)
24		uzf 9658	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
25		ffn 5431	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
26		reseq2 4959	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (ℤ≥‘𝑗)))
27		id 19	. . . . . 6 ⊢ (𝑥 = (ℤ≥‘𝑗) → 𝑥 = (ℤ≥‘𝑗))
28	26, 27	feq12d 5421	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → ((𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋))
29	28	rexrn 5724	. . . 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 11345	. . . 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 1012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm ℂ))
37	7, 36	exlimddv 1923	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
38		pmfun 6762	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 ↑pm ℂ) → Fun 𝐹)
39	37, 38	syl 14	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
40		ffvresb 5750	. . . . 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 2508	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋)))
43	41	rexbidv 2508	. . 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 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  𝒫 cpw 3617  ⟨cop 3637   class class class wbr 4047  dom cdm 4679  ran crn 4680   ↾ cres 4681  Fun wfun 5270   Fn wfn 5271  ⟶wf 5272  ‘cfv 5276  (class class class)co 5951   ↑_pm cpm 6743  ℂcc 7930  ℤcz 9379  ℤ_≥cuz 9655  TopOnctopon 14526  ⇝_𝑡clm 14703

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-pm 6745  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-inn 9044  df-n0 9303  df-z 9380  df-uz 9656  df-top 14514  df-topon 14527  df-lm 14706

This theorem is referenced by: (None)