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Theorem isperf 21249
Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
isperf (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Proof of Theorem isperf
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6379 . . . 4 (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2 unieq 4604 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 lpfval.1 . . . . 5 𝑋 = 𝐽
42, 3syl6eqr 2817 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
51, 4fveq12d 6386 . . 3 (𝑗 = 𝐽 → ((limPt‘𝑗)‘ 𝑗) = ((limPt‘𝐽)‘𝑋))
65, 4eqeq12d 2780 . 2 (𝑗 = 𝐽 → (((limPt‘𝑗)‘ 𝑗) = 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7 df-perf 21235 . 2 Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}
86, 7elrab2 3525 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155   cuni 4596  cfv 6070  Topctop 20991  limPtclp 21232  Perfcperf 21233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-iota 6033  df-fv 6078  df-perf 21235
This theorem is referenced by:  isperf2  21250  perflp  21252  perftop  21254  restperf  21282
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