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Mirrors > Home > MPE Home > Th. List > uffclsflim | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
uffclsflim | ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ufilfil 21518 | . . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋)) | |
2 | fclsfnflim 21641 | . . . . . . 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 500 | . . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∃𝑓 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓))) |
5 | simprrr 801 | . . . . . 6 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓)) | |
6 | simpll 786 | . . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ (UFil‘𝑋)) | |
7 | simprl 790 | . . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋)) | |
8 | simprrl 800 | . . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ⊆ 𝑓) | |
9 | ufilmax 21521 | . . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑓) → 𝐹 = 𝑓) | |
10 | 6, 7, 8, 9 | syl3anc 1318 | . . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 = 𝑓) |
11 | 10 | oveq2d 6565 | . . . . . 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 2691 | . . . . 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 3017 | . . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹)) |
14 | 13 | ex 449 | . . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹))) |
15 | 14 | ssrdv 3574 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹)) |
16 | flimfcls 21640 | . . 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 3585 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Filcfil 21459 UFilcufil 21513 fLim cflim 21548 fClus cfcls 21550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-fbas 19564 df-fg 19565 df-top 20521 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-fil 21460 df-ufil 21515 df-flim 21553 df-fcls 21555 |
This theorem is referenced by: ufilcmp 21646 uffcfflf 21653 |
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