Theorem fbflim 23920

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 23822	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
3	1, 2	eqeltrid 2840	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4		flimopn 23919	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
5	3, 4	sylan2 593	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
6		toponss 22871	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
7	6	ad4ant14 752	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
8	1	eleq2i 2828	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ (𝑋filGen𝐵))
9		elfg 23815	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))
10	9	ad3antlr 731	. . . . . . 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 707	. . . . 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 3157	. . 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 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358  fBascfbas 21297  filGencfg 21298  TopOnctopon 22854  Filcfil 23789   fLim cflim 23878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-ntr 22964  df-nei 23042  df-fil 23790  df-flim 23883

This theorem is referenced by:  fbflim2  23921