Theorem fclsss1 22245

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 1203	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽 ⊆ 𝐾)
2		ssralv 3885	. . . . . . 7 ⊢ (𝐽 ⊆ 𝐾 → (∀𝑜 ∈ 𝐾 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
3	2	anim2d 605	. . . . . 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 1201	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
6		fclstopon 22235	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐾 fClus 𝐹) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
7	6	adantl 475	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
8	5, 7	mpbird 249	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐾 ∈ (TopOn‘𝑋))
9		fclsopn 22237	. . . . . 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 579	. . . . 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 1199	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
12		fclsopn 22237	. . . . . 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 579	. . . . 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 286	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fClus 𝐹)) → (𝑥 ∈ (𝐾 fClus 𝐹) → 𝑥 ∈ (𝐽 fClus 𝐹)))
15	14	ex 403	. . 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 3827	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  ‘cfv 6137  (class class class)co 6924  TopOnctopon 21133  Filcfil 22068   fClus cfcls 22159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-fbas 20150  df-top 21117  df-topon 21134  df-cld 21242  df-ntr 21243  df-cls 21244  df-fil 22069  df-fcls 22164

This theorem is referenced by:  fclscf  22248