Theorem flimfcls 24144

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 24083	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2765	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimfil 24087	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimclsi 24096	. . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
5	4	sseld 3938	. . . . 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 24127	. . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
9	1, 3, 7, 8	syl3anbrc 1360	. 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
10	9	ssriv 3943	1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

Syntax hints:   ∈ wcel 2145  ∀wral 3079   ⊆ wss 3907  ∪ cuni 4868  ‘cfv 6525  (class class class)co 7400  Topctop 23011  clsccl 23136  Filcfil 23963   fLim cflim 24052   fClus cfcls 24054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-fbas 21479  df-top 23012  df-cld 23137  df-ntr 23138  df-cls 23139  df-nei 23216  df-fil 23964  df-flim 24057  df-fcls 24059

This theorem is referenced by:  fclsfnflim  24145  flimfnfcls  24146  uffclsflim  24149  flfssfcf  24156  cnpfcf  24159  cfilfcls  25394