Theorem fclssscls 23960

Description: The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclssscls	⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem fclssscls

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

Step	Hyp	Ref	Expression
1		eqid 2734	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isfcls 23951	. . . 4 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
3	2	simp3bi 1147	. . 3 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) → ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠))
4		fveq2 6832	. . . . 5 ⊢ (𝑠 = 𝑆 → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝑆))
5	4	eleq2d 2820	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘𝑠) ↔ 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
6	5	rspcv 3570	. . 3 ⊢ (𝑆 ∈ 𝐹 → (∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
7	3, 6	syl5 34	. 2 ⊢ (𝑆 ∈ 𝐹 → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
8	7	ssrdv 3937	1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ⊆ wss 3899  ∪ cuni 4861  ‘cfv 6490  (class class class)co 7356  Topctop 22835  clsccl 22960  Filcfil 23787   fClus cfcls 23878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fbas 21304  df-fil 23788  df-fcls 23883

This theorem is referenced by:  fclscmp  23972