Theorem flimss2 23876

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 2729	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimelbas 23872	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ ∪ 𝐽)
4		simpl1 1192	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 22818	. . . . . 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 2831	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ 𝑋)
8		flimneiss 23870	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
9	8	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
10		simpl3 1194	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐺 ⊆ 𝐹)
11	9, 10	sstrd 3948	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
12		simpl2 1193	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
13		elflim 23875	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
14	4, 12, 13	syl2anc 584	. . . 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 713	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ (𝐽 fLim 𝐹))
16	15	ex 412	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ (𝐽 fLim 𝐹)))
17	16	ssrdv 3943	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3905  {csn 4579  ∪ cuni 4861  ‘cfv 6486  (class class class)co 7353  TopOnctopon 22814  neicnei 23001  Filcfil 23749   fLim cflim 23838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-fbas 21277  df-top 22798  df-topon 22815  df-fil 23750  df-flim 23843

This theorem is referenced by:  flimfnfcls  23932  cnpfcf  23945