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Theorem isperf2 21177
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 21176 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
3 ssid 3773 . . . . 5 𝑋𝑋
41lpss 21167 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
53, 4mpan2 671 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
6 eqss 3767 . . . . 5 (((limPt‘𝐽)‘𝑋) = 𝑋 ↔ (((limPt‘𝐽)‘𝑋) ⊆ 𝑋𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
76baib 525 . . . 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 564 . 2 ((𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
102, 9bitri 264 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wss 3723   cuni 4575  cfv 6030  Topctop 20918  limPtclp 21159  Perfcperf 21160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-top 20919  df-cld 21044  df-cls 21046  df-lp 21161  df-perf 21162
This theorem is referenced by:  isperf3  21178
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