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Theorem isperf 22999
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 6882 . . . 4 (𝑗 = 𝐽 β†’ (limPtβ€˜π‘—) = (limPtβ€˜π½))
2 unieq 4911 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
3 lpfval.1 . . . . 5 𝑋 = βˆͺ 𝐽
42, 3eqtr4di 2782 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
51, 4fveq12d 6889 . . 3 (𝑗 = 𝐽 β†’ ((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = ((limPtβ€˜π½)β€˜π‘‹))
65, 4eqeq12d 2740 . 2 (𝑗 = 𝐽 β†’ (((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = βˆͺ 𝑗 ↔ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
7 df-perf 22985 . 2 Perf = {𝑗 ∈ Top ∣ ((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = βˆͺ 𝑗}
86, 7elrab2 3679 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4900  β€˜cfv 6534  Topctop 22739  limPtclp 22982  Perfcperf 22983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-perf 22985
This theorem is referenced by:  isperf2  23000  perflp  23002  perftop  23004  restperf  23032
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