Theorem flimss1 24033

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 2762	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
2	1	flimelbas 24028	. . . . . 6 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ ∪ 𝐾)
3	2	adantl 485	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐾)
4		simpl2 1206	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
5		filunibas 23941	. . . . . . 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 24029	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐾))
8	7	adantl 485	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘∪ 𝐾))
9		filunibas 23941	. . . . . . 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 2799	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = ∪ 𝐾)
12	3, 11	eleqtrrd 2865	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ 𝑋)
13		simpl1 1205	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
14		topontop 22973	. . . . . . 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 24025	. . . . . . 7 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐾 ∈ Top)
17	16	adantl 485	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐾 ∈ Top)
18		toponuni 22974	. . . . . . . 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 2799	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐽 = ∪ 𝐾)
21		simpl3 1207	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ⊆ 𝐾)
22		eqid 2762	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22, 1	topssnei 23184	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐾) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
24	15, 17, 20, 21, 23	syl31anc 1392	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
25		flimneiss 24026	. . . . . 6 ⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹)
26	25	adantl 485	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹)
27	24, 26	sstrd 3946	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
28		elflim 24031	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
29	13, 4, 28	syl2anc 593	. . . 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 723	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
31	30	ex 416	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
32	31	ssrdv 3942	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904  {csn 4582  ∪ cuni 4865  ‘cfv 6521  (class class class)co 7396  Topctop 22953  TopOnctopon 22970  neicnei 23157  Filcfil 23905   fLim cflim 23994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-fbas 21421  df-top 22954  df-topon 22971  df-ntr 23080  df-nei 23158  df-fil 23906  df-flim 23999

This theorem is referenced by:  flimcf  24042