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Mirrors > Home > MPE Home > Th. List > flimtop | Structured version Visualization version GIF version |
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
Ref | Expression |
---|---|
flimtop | ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | elflim2 22574 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) ∧ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) |
3 | 2 | simplbi 500 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽)) |
4 | 3 | simp1d 1138 | 1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ⊆ wss 3938 𝒫 cpw 4541 {csn 4569 ∪ cuni 4840 ran crn 5558 ‘cfv 6357 (class class class)co 7158 Topctop 21503 neicnei 21707 Filcfil 22455 fLim cflim 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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-top 21504 df-flim 22549 |
This theorem is referenced by: flimfil 22579 flimtopon 22580 flimss1 22583 flimclsi 22588 hausflimlem 22589 flimsncls 22596 cnpflfi 22609 flimfcls 22636 flimfnfcls 22638 |
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