Theorem fclsss1 24082

Description: A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclsss1	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))

Proof of Theorem fclsss1

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

Step	Hyp	Ref	Expression
1		simpl3 1207	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽 ⊆ 𝐾)
2		ssralv 4005	. . . . . . 7 ⊢ (𝐽 ⊆ 𝐾 → (∀𝑜 ∈ 𝐾 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
3	2	anim2d 621	. . . . . 6 ⊢ (𝐽 ⊆ 𝐾 → ((𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐾 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)) → (𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
4	1, 3	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → ((𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐾 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)) → (𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
5		simpl2 1206	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
6		fclstopon 24072	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐾 fClus 𝐹) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
7	6	adantl 485	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
8	5, 7	mpbird 259	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐾 ∈ (TopOn‘𝑋))
9		fclsopn 24074	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐾 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐾 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
10	8, 5, 9	syl2anc 593	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐾 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
11		simpl1 1205	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
12		fclsopn 24074	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
13	11, 5, 12	syl2anc 593	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
14	4, 10, 13	3imtr4d 296	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹)))
15	14	ex 416	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fClus 𝐹) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹))))
16	15	pm2.43d 53	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹)))
17	16	ssrdv 3942	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ‘cfv 6521  (class class class)co 7396  TopOnctopon 22970  Filcfil 23905   fClus cfcls 23996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-fbas 21421  df-top 22954  df-topon 22971  df-cld 23079  df-ntr 23080  df-cls 23081  df-fil 23906  df-fcls 24001

This theorem is referenced by:  fclscf  24085