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Theorem isperf2 21688
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 21687 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
3 ssid 3986 . . . . 5 𝑋𝑋
41lpss 21678 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
53, 4mpan2 687 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
6 eqss 3979 . . . . 5 (((limPt‘𝐽)‘𝑋) = 𝑋 ↔ (((limPt‘𝐽)‘𝑋) ⊆ 𝑋𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
76baib 536 . . . 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 575 . 2 ((𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
102, 9bitri 276 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wss 3933   cuni 4830  cfv 6348  Topctop 21429  limPtclp 21670  Perfcperf 21671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-cls 21557  df-lp 21672  df-perf 21673
This theorem is referenced by:  isperf3  21689
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