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Theorem isperf 23159
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 6906 . . . 4 (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2 unieq 4918 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 lpfval.1 . . . . 5 𝑋 = 𝐽
42, 3eqtr4di 2795 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
51, 4fveq12d 6913 . . 3 (𝑗 = 𝐽 → ((limPt‘𝑗)‘ 𝑗) = ((limPt‘𝐽)‘𝑋))
65, 4eqeq12d 2753 . 2 (𝑗 = 𝐽 → (((limPt‘𝑗)‘ 𝑗) = 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7 df-perf 23145 . 2 Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}
86, 7elrab2 3695 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108   cuni 4907  cfv 6561  Topctop 22899  limPtclp 23142  Perfcperf 23143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-perf 23145
This theorem is referenced by:  isperf2  23160  perflp  23162  perftop  23164  restperf  23192
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