Theorem uffclsflim 24060

Description: The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
uffclsflim	⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Proof of Theorem uffclsflim

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

Step	Hyp	Ref	Expression
1		ufilfil 23933	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2		fclsfnflim 24056	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓))))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓))))
4	3	biimpa 476	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∃𝑓 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))
5		simprrr 781	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓))
6		simpll 766	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ (UFil‘𝑋))
7		simprl 770	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
8		simprrl 780	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ⊆ 𝑓)
9		ufilmax 23936	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) → 𝐹 = 𝑓)
10	6, 7, 8, 9	syl3anc 1371	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 = 𝑓)
11	10	oveq2d 7464	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐽 fLim 𝐹) = (𝐽 fLim 𝑓))
12	5, 11	eleqtrrd 2847	. . . . 5 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝐹))
13	4, 12	rexlimddv 3167	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
14	13	ex 412	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
15	14	ssrdv 4014	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
16		flimfcls 24055	. . 3 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
17	16	a1i 11	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹))
18	15, 17	eqssd 4026	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ⊆ wss 3976  ‘cfv 6573  (class class class)co 7448  Filcfil 23874  UFilcufil 23928   fLim cflim 23963   fClus cfcls 23965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1o 8522  df-2o 8523  df-en 9004  df-fin 9007  df-fi 9480  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-fil 23875  df-ufil 23930  df-flim 23968  df-fcls 23970

This theorem is referenced by:  ufilcmp  24061  uffcfflf  24068