Theorem flimelbas 23831

Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
flimuni.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
flimelbas	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋)

Proof of Theorem flimelbas

Step	Hyp	Ref	Expression
1		flimuni.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	elflim2 23827	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simprbi 496	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
4	3	simpld 494	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867  ran crn 5632  ‘cfv 6499  (class class class)co 7369  Topctop 22756  neicnei 22960  Filcfil 23708   fLim cflim 23797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-top 22757  df-flim 23802

This theorem is referenced by:  flimfil  23832  flimss2  23835  flimss1  23836  flimclsi  23841  hausflimi  23843  flimsncls  23849  cnpflfi  23862  cnflf  23865  cnflf2  23866  flimcfil  25190