Theorem flimelbas 23943

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 23939	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simprbi 497	. 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 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851  ran crn 5625  ‘cfv 6492  (class class class)co 7360  Topctop 22868  neicnei 23072  Filcfil 23820   fLim cflim 23909

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-sep 5231  ax-nul 5241  ax-pr 5370

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-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-top 22869  df-flim 23914

This theorem is referenced by:  flimfil  23944  flimss2  23947  flimss1  23948  flimclsi  23953  hausflimi  23955  flimsncls  23961  cnpflfi  23974  cnflf  23977  cnflf2  23978  flimcfil  25291