Theorem flimss2 23955

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 2739	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimelbas 23951	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 482	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ ∪ 𝐽)
4		simpl1 1198	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 22897	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	4, 5	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑋 = ∪ 𝐽)
7	3, 6	eleqtrrd 2842	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ 𝑋)
8		flimneiss 23949	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
9	8	adantl 482	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
10		simpl3 1200	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐺 ⊆ 𝐹)
11	9, 10	sstrd 3925	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
12		simpl2 1199	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
13		elflim 23954	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
14	4, 12, 13	syl2anc 590	. . . 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 719	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ (𝐽 fLim 𝐹))
16	15	ex 413	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ (𝐽 fLim 𝐹)))
17	16	ssrdv 3921	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  {csn 4555  ∪ cuni 4838  ‘cfv 6485  (class class class)co 7356  TopOnctopon 22893  neicnei 23080  Filcfil 23828   fLim cflim 23917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fbas 21344  df-top 22877  df-topon 22894  df-fil 23829  df-flim 23922

This theorem is referenced by:  flimfnfcls  24011  cnpfcf  24024