Theorem flimss2 24198

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 2760	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimelbas 24194	. . . . . 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 23139	. . . . . 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 2863	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ 𝑋)
8		flimneiss 24192	. . . . . 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 3941	. . . 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 24197	. . . . 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 3937	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 2145   ⊆ wss 3899  {csn 4584  ∪ cuni 4867  ‘cfv 6533  (class class class)co 7413  TopOnctopon 23135  neicnei 23322  Filcfil 24071   fLim cflim 24160

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-fbas 21582  df-top 23119  df-topon 23136  df-fil 24072  df-flim 24165

This theorem is used by:  flimfnfcls  24254  cnpfcf  24267