Theorem fbflim 21590

Description: A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
fbflim.3	⊢ 𝐹 = (𝑋filGen𝐵)

Assertion

Ref	Expression
fbflim	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦   𝑥,𝐹,𝑦

Proof of Theorem fbflim

Step	Hyp	Ref	Expression
1		fbflim.3	. . . 4 ⊢ 𝐹 = (𝑋filGen𝐵)
2		fgcl 21492	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
3	1, 2	syl5eqel 2692	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		flimopn 21589	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
5	3, 4	sylan2 490	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
6	1	eleq2i 2680	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ (𝑋filGen𝐵))
7		elfg 21485	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
8	7	ad3antlr 763	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
9	6, 8	syl5bb 271	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
10		simpll 786	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11		toponss 20544	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
12	10, 11	sylan 487	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
13	12	biantrurd 528	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
14	9, 13	bitr4d 270	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐹 ↔ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))
15	14	imbi2d 329	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → ((𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) ↔ (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
16	15	ralbidva 2968	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
17	16	pm5.32da 671	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))))
18	5, 17	bitrd 267	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  fBascfbas 19555  filGencfg 19556  TopOnctopon 20518  Filcfil 21459   fLim cflim 21548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fg 19565  df-top 20521  df-topon 20523  df-ntr 20634  df-nei 20712  df-fil 21460  df-flim 21553

This theorem is referenced by:  fbflim2  21591