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Theorem isperf2 21845
 Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
isperf2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))

Proof of Theorem isperf2
StepHypRef Expression
1 lpfval.1 . . 3 𝑋 = 𝐽
21isperf 21844 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
3 ssid 3915 . . . . 5 𝑋𝑋
41lpss 21835 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
53, 4mpan2 691 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
6 eqss 3908 . . . . 5 (((limPt‘𝐽)‘𝑋) = 𝑋 ↔ (((limPt‘𝐽)‘𝑋) ⊆ 𝑋𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
76baib 540 . . . 4 (((limPt‘𝐽)‘𝑋) ⊆ 𝑋 → (((limPt‘𝐽)‘𝑋) = 𝑋𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
85, 7syl 17 . . 3 (𝐽 ∈ Top → (((limPt‘𝐽)‘𝑋) = 𝑋𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
98pm5.32i 579 . 2 ((𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
102, 9bitri 278 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859  ∪ cuni 4799  ‘cfv 6336  Topctop 21586  limPtclp 21827  Perfcperf 21828 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-top 21587  df-cld 21712  df-cls 21714  df-lp 21829  df-perf 21830 This theorem is referenced by:  isperf3  21846
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