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

Theorem isperf 23068
Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
isperf (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))

Proof of Theorem isperf
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6897 . . . 4 (𝑗 = 𝐽 β†’ (limPtβ€˜π‘—) = (limPtβ€˜π½))
2 unieq 4919 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
3 lpfval.1 . . . . 5 𝑋 = βˆͺ 𝐽
42, 3eqtr4di 2786 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
51, 4fveq12d 6904 . . 3 (𝑗 = 𝐽 β†’ ((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = ((limPtβ€˜π½)β€˜π‘‹))
65, 4eqeq12d 2744 . 2 (𝑗 = 𝐽 β†’ (((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = βˆͺ 𝑗 ↔ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
7 df-perf 23054 . 2 Perf = {𝑗 ∈ Top ∣ ((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = βˆͺ 𝑗}
86, 7elrab2 3685 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆͺ cuni 4908  β€˜cfv 6548  Topctop 22808  limPtclp 23051  Perfcperf 23052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-perf 23054
This theorem is referenced by:  isperf2  23069  perflp  23071  perftop  23073  restperf  23101
  Copyright terms: Public domain W3C validator