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Theorem isperf 20765
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 6103 . . . 4 (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2 unieq 4380 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 lpfval.1 . . . . 5 𝑋 = 𝐽
42, 3syl6eqr 2662 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
51, 4fveq12d 6109 . . 3 (𝑗 = 𝐽 → ((limPt‘𝑗)‘ 𝑗) = ((limPt‘𝐽)‘𝑋))
65, 4eqeq12d 2625 . 2 (𝑗 = 𝐽 → (((limPt‘𝑗)‘ 𝑗) = 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7 df-perf 20751 . 2 Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}
86, 7elrab2 3333 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977   cuni 4372  cfv 5804  Topctop 20517  limPtclp 20748  Perfcperf 20749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-perf 20751
This theorem is referenced by:  isperf2  20766  perflp  20768  perftop  20770  restperf  20798
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