Theorem fclsss2 23969

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 4001	. . . . . 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 23958	. . . . . . . 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 23959	. . . . . 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 23959	. . . . . 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 3938	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 3050   ⊆ wss 3900  ‘cfv 6491  (class class class)co 7358  TopOnctopon 22856  clsccl 22964  Filcfil 23791   fClus cfcls 23882

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fbas 21308  df-topon 22857  df-fil 23792  df-fcls 23887

This theorem is referenced by:  fclsfnflim  23973  ufilcmp  23978  cnpfcfi  23986