Theorem lmflf 24034

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 12906	. . . . . . . 8 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 6747	. . . . . . . 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 12924	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	eqsstri 4043	. . . . . . 7 ⊢ 𝑍 ⊆ ℤ
7		imaeq2 6085	. . . . . . . . 9 ⊢ (𝑦 = (ℤ≥‘𝑗) → (𝐹 “ 𝑦) = (𝐹 “ (ℤ≥‘𝑗)))
8	7	sseq1d 4040	. . . . . . . 8 ⊢ (𝑦 = (ℤ≥‘𝑗) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
9	8	rexima 7275	. . . . . . 7 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ) → (∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
10	3, 6, 9	mp2an 691	. . . . . 6 ⊢ (∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥)
11		simpl3 1193	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶𝑋)
12	11	ffund 6751	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → Fun 𝐹)
13		uzss 12926	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑗) ⊆ (ℤ≥‘𝑀))
14	13, 4	eleq2s 2862	. . . . . . . . . 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 6757	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = 𝑍)
17	16, 4	eqtrdi 2796	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = (ℤ≥‘𝑀))
18	15, 17	sseqtrrd 4050	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ dom 𝐹)
19		funimass4 6986	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
20	12, 18, 19	syl2anc 583	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
21	20	rexbidva 3183	. . . . . 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 3184	. . 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 629	. 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 2741	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
30	26, 4, 27, 28, 29	lmbrf 23289	. 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 23927	. . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍))
32		lmflf.2	. . . 4 ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))
33	32	flffbas 24024	. . 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166  dom cdm 5700   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℤcz 12639  ℤ_≥cuz 12903  fBascfbas 21375  filGencfg 21376  TopOnctopon 22937  ⇝_𝑡clm 23255   fLimf cflf 23964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-i2m1 11252  ax-1ne0 11253  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-neg 11523  df-nn 12294  df-z 12640  df-uz 12904  df-rest 17482  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-ntr 23049  df-nei 23127  df-lm 23258  df-fil 23875  df-fm 23967  df-flim 23968  df-flf 23969

This theorem is referenced by:  cmetcaulem  25341