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Mirrors > Home > MPE Home > Th. List > isperf2 | Structured version Visualization version GIF version |
Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
isperf2 | ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpfval.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | isperf 21753 | . 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
3 | ssid 3989 | . . . . 5 ⊢ 𝑋 ⊆ 𝑋 | |
4 | 1 | lpss 21744 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋) |
5 | 3, 4 | mpan2 689 | . . . 4 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋) |
6 | eqss 3982 | . . . . 5 ⊢ (((limPt‘𝐽)‘𝑋) = 𝑋 ↔ (((limPt‘𝐽)‘𝑋) ⊆ 𝑋 ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋))) | |
7 | 6 | baib 538 | . . . 4 ⊢ (((limPt‘𝐽)‘𝑋) ⊆ 𝑋 → (((limPt‘𝐽)‘𝑋) = 𝑋 ↔ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋))) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝐽 ∈ Top → (((limPt‘𝐽)‘𝑋) = 𝑋 ↔ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋))) |
9 | 8 | pm5.32i 577 | . 2 ⊢ ((𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋))) |
10 | 2, 9 | bitri 277 | 1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ∪ cuni 4832 ‘cfv 6350 Topctop 21495 limPtclp 21736 Perfcperf 21737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-top 21496 df-cld 21621 df-cls 21623 df-lp 21738 df-perf 21739 |
This theorem is referenced by: isperf3 21755 |
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