Theorem flimfcls 24164

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 24103	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2763	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimfil 24107	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimclsi 24116	. . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
5	4	sseld 3937	. . . . 5 ⊢ (𝑥 ∈ 𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
6	5	com12 33	. . . 4 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ 𝐹 → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
7	6	ralrimiv 3156	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
8	2	isfcls 24147	. . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
9	1, 3, 7, 8	syl3anbrc 1362	. 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
10	9	ssriv 3942	1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

Syntax hints:   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906  ∪ cuni 4873  ‘cfv 6538  (class class class)co 7412  Topctop 23031  clsccl 23156  Filcfil 23983   fLim cflim 24072   fClus cfcls 24074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fbas 21500  df-top 23032  df-cld 23157  df-ntr 23158  df-cls 23159  df-nei 23236  df-fil 23984  df-flim 24077  df-fcls 24079

This theorem is referenced by:  fclsfnflim  24165  flimfnfcls  24166  uffclsflim  24169  flfssfcf  24176  cnpfcf  24179  cfilfcls  25414