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Theorem isperf 22486
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 6839 . . . 4 (𝑗 = 𝐽 β†’ (limPtβ€˜π‘—) = (limPtβ€˜π½))
2 unieq 4874 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
3 lpfval.1 . . . . 5 𝑋 = βˆͺ 𝐽
42, 3eqtr4di 2794 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
51, 4fveq12d 6846 . . 3 (𝑗 = 𝐽 β†’ ((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = ((limPtβ€˜π½)β€˜π‘‹))
65, 4eqeq12d 2752 . 2 (𝑗 = 𝐽 β†’ (((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = βˆͺ 𝑗 ↔ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
7 df-perf 22472 . 2 Perf = {𝑗 ∈ Top ∣ ((limPtβ€˜π‘—)β€˜βˆͺ 𝑗) = βˆͺ 𝑗}
86, 7elrab2 3646 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆͺ cuni 4863  β€˜cfv 6493  Topctop 22226  limPtclp 22469  Perfcperf 22470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-perf 22472
This theorem is referenced by:  isperf2  22487  perflp  22489  perftop  22491  restperf  22519
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