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Mirrors > Home > MPE Home > Th. List > perflp | Structured version Visualization version GIF version |
Description: The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
perflp | ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpfval.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | isperf 23174 | . 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
3 | 2 | simprbi 496 | 1 ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ∪ cuni 4911 ‘cfv 6562 Topctop 22914 limPtclp 23157 Perfcperf 23158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-perf 23160 |
This theorem is referenced by: (None) |
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