MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isperf Structured version   Visualization version   GIF version

Theorem isperf 23174
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 4922 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 lpfval.1 . . . . 5 𝑋 = 𝐽
42, 3eqtr4di 2792 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
51, 4fveq12d 6913 . . 3 (𝑗 = 𝐽 → ((limPt‘𝑗)‘ 𝑗) = ((limPt‘𝐽)‘𝑋))
65, 4eqeq12d 2750 . 2 (𝑗 = 𝐽 → (((limPt‘𝑗)‘ 𝑗) = 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7 df-perf 23160 . 2 Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}
86, 7elrab2 3697 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105   cuni 4911  cfv 6562  Topctop 22914  limPtclp 23157  Perfcperf 23158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-perf 23160
This theorem is referenced by:  isperf2  23175  perflp  23177  perftop  23179  restperf  23207
  Copyright terms: Public domain W3C validator