Theorem fclsss2 24342

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 1212	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ⊆ 𝐺)
2		ssralv 3977	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → (∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠) → ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
3	1, 2	syl 18	. . . . 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 1210	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		fclstopon 24331	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐽 fClus 𝐺) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐺 ∈ (Fil‘𝑋)))
6	5	adantl 487	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐺 ∈ (Fil‘𝑋)))
7	4, 6	mpbid 235	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐺 ∈ (Fil‘𝑋))
8		isfcls2 24332	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐺) ↔ ∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
9	4, 7, 8	syl2anc 596	. . . . 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 1211	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
11		isfcls2 24332	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
12	4, 10, 11	syl2anc 596	. . . . 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 297	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹)))
14	13	ex 418	. . 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 54	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑥 ∈ (𝐽 fClus 𝐺) → 𝑥 ∈ (𝐽 fClus 𝐹)))
16	15	ssrdv 3914	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  TopOnctopon 23228  clsccl 23336  Filcfil 24164   fClus cfcls 24255

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-fbas 21675  df-topon 23229  df-fil 24165  df-fcls 24260

This theorem is used by:  fclsfnflim  24346  ufilcmp  24351  cnpfcfi  24359