Theorem fclsss2 23971

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 1195	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ⊆ 𝐺)
2		ssralv 4003	. . . . . 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 1193	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐽 ∈ (TopOn‘𝑋))
5		fclstopon 23960	. . . . . . . 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 23961	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐺) ↔ ∀𝑠 ∈ 𝐺 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
9	4, 7, 8	syl2anc 585	. . . . 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 1194	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) ∧ 𝑥 ∈ (𝐽 fClus 𝐺)) → 𝐹 ∈ (Fil‘𝑋))
11		isfcls2 23961	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
12	4, 10, 11	syl2anc 585	. . . . 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 3940	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  ‘cfv 6493  (class class class)co 7360  TopOnctopon 22858  clsccl 22966  Filcfil 23793   fClus cfcls 23884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21310  df-topon 22859  df-fil 23794  df-fcls 23889

This theorem is referenced by:  fclsfnflim  23975  ufilcmp  23980  cnpfcfi  23988