Theorem flimss1 23032

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 2738	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
2	1	flimelbas 23027	. . . . . 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 1190	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
5		filunibas 22940	. . . . . . 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 23028	. . . . . . . 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 22940	. . . . . . 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 2780	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = ∪ 𝐾)
12	3, 11	eleqtrrd 2842	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ 𝑋)
13		simpl1 1189	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
14		topontop 21970	. . . . . . 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 23024	. . . . . . 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 21971	. . . . . . . 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 2780	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐽 = ∪ 𝐾)
21		simpl3 1191	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ⊆ 𝐾)
22		eqid 2738	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22, 1	topssnei 22183	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐾) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
24	15, 17, 20, 21, 23	syl31anc 1371	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
25		flimneiss 23025	. . . . . 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 3927	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
28		elflim 23030	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
29	13, 4, 28	syl2anc 583	. . . 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 709	. . 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 3923	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  {csn 4558  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255  Topctop 21950  TopOnctopon 21967  neicnei 22156  Filcfil 22904   fLim cflim 22993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-top 21951  df-topon 21968  df-ntr 22079  df-nei 22157  df-fil 22905  df-flim 22998

This theorem is referenced by:  flimcf  23041