Theorem flimneiss 24297

Description: A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
flimneiss	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)

Proof of Theorem flimneiss

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	elflim2 24295	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) ∧ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
3	2	simprbi 503	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
4	3	simprd 501	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146   ⊆ wss 3899  𝒫 cpw 4559  {csn 4587  ∪ cuni 4869  ran crn 5655  ‘cfv 6546  (class class class)co 7427  Topctop 23223  neicnei 23427  Filcfil 24176   fLim cflim 24265

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-top 23224  df-flim 24270

This theorem is used by:  flimnei  24298  flimfil  24300  flimss2  24303  flimss1  24304  flimcf  24313