Theorem isflf 23977

Description: The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Assertion

Ref	Expression
isflf	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))

Distinct variable groups:   𝐴,𝑜   𝑜,𝑠,𝐹   𝑜,𝐽,𝑠   𝑜,𝐿,𝑠   𝑜,𝑋,𝑠   𝑜,𝑌,𝑠

Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem isflf

Step	Hyp	Ref	Expression
1		flfval 23974	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	eleq2d 2825	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
3		simp1 1142	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		toponmax 22910	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5	4	3ad2ant1 1139	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐽)
6		filfbas 23832	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
7	6	3ad2ant2 1140	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐿 ∈ (fBas‘𝑌))
8		simp3 1144	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
9		fmfil 23928	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
10	5, 7, 8, 9	syl3anc 1379	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
11		flimopn 23959	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))))
12	3, 10, 11	syl2anc 590	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))))
13		toponss 22911	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
14	3, 13	sylan 586	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
15		elfm 23931	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
16	5, 7, 8, 15	syl3anc 1379	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
17	16	adantr 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
18	14, 17	mpbirand 713	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
19	18	imbi2d 341	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → ((𝐴 ∈ 𝑜 → 𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
20	19	ralbidva 3160	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
21	20	anbi2d 636	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿))) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))
22	2, 12, 21	3bitrd 306	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883   “ cima 5622  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  fBascfbas 21336  TopOnctopon 22894  Filcfil 23829   FilMap cfm 23917   fLim cflim 23918   fLimf cflf 23919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8766  df-fbas 21345  df-fg 21346  df-top 22878  df-topon 22895  df-ntr 23004  df-nei 23082  df-fil 23830  df-fm 23922  df-flim 23923  df-flf 23924

This theorem is referenced by:  flfelbas  23978  flffbas  23979  flftg  23980  cnpflfi  23983  cnpflf2  23984  txflf  23990  limcflf  25867  rrhre  34214