Theorem lmflf 24029

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 12879	. . . . . . . 8 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 6737	. . . . . . . 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 12897	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	eqsstri 4030	. . . . . . 7 ⊢ 𝑍 ⊆ ℤ
7		imaeq2 6076	. . . . . . . . 9 ⊢ (𝑦 = (ℤ≥‘𝑗) → (𝐹 “ 𝑦) = (𝐹 “ (ℤ≥‘𝑗)))
8	7	sseq1d 4027	. . . . . . . 8 ⊢ (𝑦 = (ℤ≥‘𝑗) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
9	8	rexima 7258	. . . . . . 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 1192	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶𝑋)
12	11	ffund 6741	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → Fun 𝐹)
13		uzss 12899	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑗) ⊆ (ℤ≥‘𝑀))
14	13, 4	eleq2s 2857	. . . . . . . . . 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 6747	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = 𝑍)
17	16, 4	eqtrdi 2791	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = (ℤ≥‘𝑀))
18	15, 17	sseqtrrd 4037	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ dom 𝐹)
19		funimass4 6973	. . . . . . . 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 3175	. . . . . 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 3176	. . 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 1135	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		simp2 1136	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝑀 ∈ ℤ)
28		simp3 1137	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐹:𝑍⟶𝑋)
29		eqidd 2736	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
30	26, 4, 27, 28, 29	lmbrf 23284	. 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 23922	. . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍))
32		lmflf.2	. . . 4 ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))
33	32	flffbas 24019	. . 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 1163	. 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689   “ cima 5692  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℤcz 12611  ℤ_≥cuz 12876  fBascfbas 21370  filGencfg 21371  TopOnctopon 22932  ⇝_𝑡clm 23250   fLimf cflf 23959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-i2m1 11221  ax-1ne0 11222  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-neg 11493  df-nn 12265  df-z 12612  df-uz 12877  df-rest 17469  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-lm 23253  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964

This theorem is referenced by:  cmetcaulem  25336