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Theorem isperf 21687
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 6663 . . . 4 (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2 unieq 4838 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 lpfval.1 . . . . 5 𝑋 = 𝐽
42, 3syl6eqr 2871 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
51, 4fveq12d 6670 . . 3 (𝑗 = 𝐽 → ((limPt‘𝑗)‘ 𝑗) = ((limPt‘𝐽)‘𝑋))
65, 4eqeq12d 2834 . 2 (𝑗 = 𝐽 → (((limPt‘𝑗)‘ 𝑗) = 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7 df-perf 21673 . 2 Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}
86, 7elrab2 3680 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105   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
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-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-perf 21673
This theorem is referenced by:  isperf2  21688  perflp  21690  perftop  21692  restperf  21720
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