Theorem fclsss2 24032

Description: A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
fclsss2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

Proof of Theorem fclsss2

Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl3 1193	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ⊆ 𝐺)
2		ssralv 4051	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → (∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠) → ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
3	1, 2	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠) → ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
4		simpl1 1191	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		fclstopon 24021	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐽 fClus 𝐺) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐺 ∈ (Fil‘𝑋)))
6	5	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐺 ∈ (Fil‘𝑋)))
7	4, 6	mpbid 232	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐺 ∈ (Fil‘𝑋))
8		isfcls2 24022	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐺) ↔ ∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
9	4, 7, 8	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐺) ↔ ∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
10		simpl2 1192	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
11		isfcls2 24022	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
12	4, 10, 11	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
13	3, 9, 12	3imtr4d 294	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹)))
14	13	ex 412	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹))))
15	14	pm2.43d 53	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹)))
16	15	ssrdv 3988	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2107  ∀wral 3060   ⊆ wss 3950  ‘cfv 6560  (class class class)co 7432  TopOnctopon 22917  clsccl 23027  Filcfil 23854   fClus cfcls 23945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-topon 22918  df-fil 23855  df-fcls 23950

This theorem is referenced by:  fclsfnflim  24036  ufilcmp  24041  cnpfcfi  24049