Theorem fbflim 23932

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 23834	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
3	1, 2	eqeltrid 2841	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		flimopn 23931	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
5	3, 4	sylan2 594	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
6		toponss 22883	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
7	6	ad4ant14 753	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
8	1	eleq2i 2829	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ (𝑋filGen𝐵))
9		elfg 23827	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
10	9	ad3antlr 732	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
11	8, 10	bitrid 283	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
12	7, 11	mpbirand 708	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐹 ↔ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))
13	12	imbi2d 340	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → ((𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) ↔ (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
14	13	ralbidva 3159	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
15	14	pm5.32da 579	. 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))))
16	5, 15	bitrd 279	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  ‘cfv 6500  (class class class)co 7368  fBascfbas 21309  filGencfg 21310  TopOnctopon 22866  Filcfil 23801   fLim cflim 23890

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-fbas 21318  df-fg 21319  df-top 22850  df-topon 22867  df-ntr 22976  df-nei 23054  df-fil 23802  df-flim 23895

This theorem is referenced by:  fbflim2  23933