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Mirrors > Home > MPE Home > Th. List > flimfcls | Structured version Visualization version GIF version |
Description: A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
flimfcls | ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flimtop 22181 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
2 | eqid 2778 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | flimfil 22185 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
4 | flimclsi 22194 | . . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥)) | |
5 | 4 | sseld 3820 | . . . . 5 ⊢ (𝑥 ∈ 𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
6 | 5 | com12 32 | . . . 4 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ 𝐹 → 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
7 | 6 | ralrimiv 3147 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)) |
8 | 2 | isfcls 22225 | . . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
9 | 1, 3, 7, 8 | syl3anbrc 1400 | . 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹)) |
10 | 9 | ssriv 3825 | 1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∀wral 3090 ⊆ wss 3792 ∪ cuni 4673 ‘cfv 6137 (class class class)co 6924 Topctop 21109 clsccl 21234 Filcfil 22061 fLim cflim 22150 fClus cfcls 22152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-fbas 20143 df-top 21110 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-fil 22062 df-flim 22155 df-fcls 22157 |
This theorem is referenced by: fclsfnflim 22243 flimfnfcls 22244 uffclsflim 22247 flfssfcf 22254 cnpfcf 22257 cfilfcls 23484 |
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