Theorem uffclsflim 24079

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 23952	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2		fclsfnflim 24075	. . . . . . 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 480	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∃𝑓 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))
5		simprrr 791	. . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓))
6		simpll 776	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ (UFil‘𝑋))
7		simprl 780	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
8		simprrl 790	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ⊆ 𝑓)
9		ufilmax 23955	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) → 𝐹 = 𝑓)
10	6, 7, 8, 9	syl3anc 1389	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 = 𝑓)
11	10	oveq2d 7407	. . . . . 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 2864	. . . . 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 3168	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
14	13	ex 416	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
15	14	ssrdv 3940	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
16		flimfcls 24074	. . 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 3951	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3902  ‘cfv 6516  (class class class)co 7391  Filcfil 23893  UFilcufil 23947   fLim cflim 23982   fClus cfcls 23984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1o 8431  df-2o 8432  df-en 8922  df-fin 8925  df-fi 9351  df-fbas 21409  df-fg 21410  df-top 22942  df-topon 22959  df-cld 23067  df-ntr 23068  df-cls 23069  df-nei 23146  df-fil 23894  df-ufil 23949  df-flim 23987  df-fcls 23989

This theorem is referenced by:  ufilcmp  24080  uffcfflf  24087