Theorem flimss1 23916

Description: A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
flimss1	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))

Proof of Theorem flimss1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
2	1	flimelbas 23911	. . . . . 6 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ ∪ 𝐾)
3	2	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐾)
4		simpl2 1193	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
5		filunibas 23824	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
6	4, 5	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐹 = 𝑋)
7	1	flimfil 23912	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐾))
8	7	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘∪ 𝐾))
9		filunibas 23824	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘∪ 𝐾) → ∪ 𝐹 = ∪ 𝐾)
10	8, 9	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐹 = ∪ 𝐾)
11	6, 10	eqtr3d 2773	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = ∪ 𝐾)
12	3, 11	eleqtrrd 2838	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ 𝑋)
13		simpl1 1192	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
14		topontop 22856	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15	13, 14	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ Top)
16		flimtop 23908	. . . . . . 7 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐾 ∈ Top)
17	16	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐾 ∈ Top)
18		toponuni 22857	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
19	13, 18	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = ∪ 𝐽)
20	19, 11	eqtr3d 2773	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐽 = ∪ 𝐾)
21		simpl3 1194	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ⊆ 𝐾)
22		eqid 2736	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22, 1	topssnei 23067	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐾) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
24	15, 17, 20, 21, 23	syl31anc 1375	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
25		flimneiss 23909	. . . . . 6 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹)
26	25	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹)
27	24, 26	sstrd 3974	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
28		elflim 23914	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
29	13, 4, 28	syl2anc 584	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
30	12, 27, 29	mpbir2and 713	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
31	30	ex 412	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
32	31	ssrdv 3969	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3931  {csn 4606  ∪ cuni 4888  ‘cfv 6536  (class class class)co 7410  Topctop 22836  TopOnctopon 22853  neicnei 23040  Filcfil 23788   fLim cflim 23877

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-fbas 21317  df-top 22837  df-topon 22854  df-ntr 22963  df-nei 23041  df-fil 23789  df-flim 23882

This theorem is referenced by:  flimcf  23925