Theorem lmflf 23970

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 12791	. . . . . . . 8 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 6668	. . . . . . . 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 12809	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	eqsstri 3968	. . . . . . 7 ⊢ 𝑍 ⊆ ℤ
7		imaeq2 6021	. . . . . . . . 9 ⊢ (𝑦 = (ℤ≥‘𝑗) → (𝐹 “ 𝑦) = (𝐹 “ (ℤ≥‘𝑗)))
8	7	sseq1d 3953	. . . . . . . 8 ⊢ (𝑦 = (ℤ≥‘𝑗) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
9	8	rexima 7193	. . . . . . 7 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ) → (∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥))
10	3, 6, 9	mp2an 693	. . . . . 6 ⊢ (∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥)
11		simpl3 1195	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶𝑋)
12	11	ffund 6672	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → Fun 𝐹)
13		uzss 12811	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑗) ⊆ (ℤ≥‘𝑀))
14	13, 4	eleq2s 2854	. . . . . . . . . 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 6678	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = 𝑍)
17	16, 4	eqtrdi 2787	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = (ℤ≥‘𝑀))
18	15, 17	sseqtrrd 3959	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ dom 𝐹)
19		funimass4 6904	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
20	12, 18, 19	syl2anc 585	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥))
21	20	rexbidva 3159	. . . . . 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 3160	. . 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 631	. 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 1137	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		simp2 1138	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝑀 ∈ ℤ)
28		simp3 1139	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐹:𝑍⟶𝑋)
29		eqidd 2737	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
30	26, 4, 27, 28, 29	lmbrf 23225	. 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 23863	. . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍))
32		lmflf.2	. . . 4 ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))
33	32	flffbas 23960	. . 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 1165	. 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 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889  𝒫 cpw 4541   class class class wbr 5085  dom cdm 5631   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℤcz 12524  ℤ_≥cuz 12788  fBascfbas 21340  filGencfg 21341  TopOnctopon 22875  ⇝_𝑡clm 23191   fLimf cflf 23900

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-i2m1 11106  ax-1ne0 11107  ax-rrecex 11110  ax-cnre 11111  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-neg 11380  df-nn 12175  df-z 12525  df-uz 12789  df-rest 17385  df-fbas 21349  df-fg 21350  df-top 22859  df-topon 22876  df-ntr 22985  df-nei 23063  df-lm 23194  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905

This theorem is referenced by:  cmetcaulem  25255