Theorem lmflf 23921

Description: The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
lmflf.1	⊢ 𝑍 = (ℤ≥‘𝑀)
lmflf.2	⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))

Assertion

Ref	Expression
lmflf	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))

Proof of Theorem lmflf

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

Step	Hyp	Ref	Expression
1		uzf 12735	. . . . . . . 8 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 6651	. . . . . . . 8 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ ℤ≥ Fn ℤ
4		lmflf.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		uzssz 12753	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	eqsstri 3981	. . . . . . 7 ⊢ 𝑍 ⊆ ℤ
7		imaeq2 6005	. . . . . . . . 9 ⊢ (𝑦 = (ℤ≥‘𝑗) → (𝐹 “ 𝑦) = (𝐹 “ (ℤ≥‘𝑗)))
8	7	sseq1d 3966	. . . . . . . 8 ⊢ (𝑦 = (ℤ≥‘𝑗) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
9	8	rexima 7172	. . . . . . 7 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ) → (∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
10	3, 6, 9	mp2an 692	. . . . . 6 ⊢ (∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥)
11		simpl3 1194	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶𝑋)
12	11	ffund 6655	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → Fun 𝐹)
13		uzss 12755	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑗) ⊆ (ℤ≥‘𝑀))
14	13, 4	eleq2s 2849	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → (ℤ≥‘𝑗) ⊆ (ℤ≥‘𝑀))
15	14	adantl 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ (ℤ≥‘𝑀))
16	11	fdmd 6661	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = 𝑍)
17	16, 4	eqtrdi 2782	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = (ℤ≥‘𝑀))
18	15, 17	sseqtrrd 3972	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ dom 𝐹)
19		funimass4 6886	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
20	12, 18, 19	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
21	20	rexbidva 3154	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
22	10, 21	bitr2id 284	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥 ↔ ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥))
23	22	imbi2d 340	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → ((𝑃 ∈ 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥) ↔ (𝑃 ∈ 𝑥 → ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥)))
24	23	ralbidv 3155	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥)))
25	24	anbi2d 630	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → ((𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥)) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥))))
26		simp1 1136	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		simp2 1137	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝑀 ∈ ℤ)
28		simp3 1138	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐹:𝑍⟶𝑋)
29		eqidd 2732	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
30	26, 4, 27, 28, 29	lmbrf 23176	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))))
31	4	uzfbas 23814	. . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍))
32		lmflf.2	. . . 4 ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))
33	32	flffbas 23911	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍) ∧ 𝐹:𝑍⟶𝑋) → (𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥))))
34	31, 33	syl3an2 1164	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥))))
35	25, 30, 34	3bitr4d 311	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3902  𝒫 cpw 4550   class class class wbr 5091  dom cdm 5616   “ cima 5619  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℤcz 12468  ℤ_≥cuz 12732  fBascfbas 21280  filGencfg 21281  TopOnctopon 22826  ⇝_𝑡clm 23142   fLimf cflf 23851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-i2m1 11074  ax-1ne0 11075  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-neg 11347  df-nn 12126  df-z 12469  df-uz 12733  df-rest 17326  df-fbas 21289  df-fg 21290  df-top 22810  df-topon 22827  df-ntr 22936  df-nei 23014  df-lm 23145  df-fil 23762  df-fm 23854  df-flim 23855  df-flf 23856

This theorem is referenced by:  cmetcaulem  25216