Theorem flimss2 24346

Description: A limit point of a filter is a limit point of a finer filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
flimss2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹))

Proof of Theorem flimss2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimelbas 24342	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 487	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ ∪ 𝐽)
4		simpl1 1210	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 23287	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	4, 5	syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑋 = ∪ 𝐽)
7	3, 6	eleqtrrd 2867	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ 𝑋)
8		flimneiss 24340	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
9	8	adantl 487	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
10		simpl3 1212	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐺 ⊆ 𝐹)
11	9, 10	sstrd 3923	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
12		simpl2 1211	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
13		elflim 24345	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
14	4, 12, 13	syl2anc 596	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
15	7, 11, 14	mpbir2and 726	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ (𝐽 fLim 𝐹))
16	15	ex 418	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ (𝐽 fLim 𝐹)))
17	16	ssrdv 3919	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  {csn 4588  ∪ cuni 4872  ‘cfv 6566  (class class class)co 7451  TopOnctopon 23283  neicnei 23470  Filcfil 24219   fLim cflim 24308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-fbas 21726  df-top 23267  df-topon 23284  df-fil 24220  df-flim 24313

This theorem is used by:  flimfnfcls  24402  cnpfcf  24415