Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22573 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
2 | eqid 2821 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | flimfil 22577 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
4 | flimclsi 22586 | . . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥)) | |
5 | 4 | sseld 3966 | . . . . 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 3181 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)) |
8 | 2 | isfcls 22617 | . . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
9 | 1, 3, 7, 8 | syl3anbrc 1339 | . 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹)) |
10 | 9 | ssriv 3971 | 1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ∪ cuni 4838 ‘cfv 6355 (class class class)co 7156 Topctop 21501 clsccl 21626 Filcfil 22453 fLim cflim 22542 fClus cfcls 22544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-fbas 20542 df-top 21502 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-fil 22454 df-flim 22547 df-fcls 22549 |
This theorem is referenced by: fclsfnflim 22635 flimfnfcls 22636 uffclsflim 22639 flfssfcf 22646 cnpfcf 22649 cfilfcls 23877 |
Copyright terms: Public domain | W3C validator |