Theorem flimfcls 23969

Description: A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
flimfcls	⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

Proof of Theorem flimfcls

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

Step	Hyp	Ref	Expression
1		flimtop 23908	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2736	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimfil 23912	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimclsi 23921	. . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
5	4	sseld 3962	. . . . 5 ⊢ (𝑥 ∈ 𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
6	5	com12 32	. . . 4 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ 𝐹 → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
7	6	ralrimiv 3132	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
8	2	isfcls 23952	. . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
9	1, 3, 7, 8	syl3anbrc 1344	. 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
10	9	ssriv 3967	1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

Syntax hints:   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  ∪ cuni 4888  ‘cfv 6536  (class class class)co 7410  Topctop 22836  clsccl 22961  Filcfil 23788   fLim cflim 23877   fClus cfcls 23879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-fbas 21317  df-top 22837  df-cld 22962  df-ntr 22963  df-cls 22964  df-nei 23041  df-fil 23789  df-flim 23882  df-fcls 23884

This theorem is referenced by:  fclsfnflim  23970  flimfnfcls  23971  uffclsflim  23974  flfssfcf  23981  cnpfcf  23984  cfilfcls  25231