Theorem flimss2 24284

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 2761	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimelbas 24280	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 487	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ ∪ 𝐽)
4		simpl1 1210	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 23225	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	4, 5	syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑋 = ∪ 𝐽)
7	3, 6	eleqtrrd 2864	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ 𝑋)
8		flimneiss 24278	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐺) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
9	8	adantl 487	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐺)
10		simpl3 1212	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐺 ⊆ 𝐹)
11	9, 10	sstrd 3941	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
12		simpl2 1211	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
13		elflim 24283	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
14	4, 12, 13	syl2anc 596	. . . 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 726	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) ∧ 𝑥 ∈ (𝐽 fLim 𝐺)) → 𝑥 ∈ (𝐽 fLim 𝐹))
16	15	ex 418	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝑥 ∈ (𝐽 fLim 𝐺) → 𝑥 ∈ (𝐽 fLim 𝐹)))
17	16	ssrdv 3937	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  {csn 4584  ∪ cuni 4867  ‘cfv 6537  (class class class)co 7418  TopOnctopon 23221  neicnei 23408  Filcfil 24157   fLim cflim 24246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-fbas 21668  df-top 23205  df-topon 23222  df-fil 24158  df-flim 24251

This theorem is used by:  flimfnfcls  24340  cnpfcf  24353