Theorem isperf 22210

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.

Step	Hyp	Ref	Expression
1		fveq2 6756	. . . 4 ⊢ (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2		unieq 4847	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		lpfval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2797	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
5	1, 4	fveq12d 6763	. . 3 ⊢ (𝑗 = 𝐽 → ((limPt‘𝑗)‘∪ 𝑗) = ((limPt‘𝐽)‘𝑋))
6	5, 4	eqeq12d 2754	. 2 ⊢ (𝑗 = 𝐽 → (((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7		df-perf 22196	. 2 ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
8	6, 7	elrab2 3620	1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  limPtclp 22193  Perfcperf 22194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-perf 22196

This theorem is referenced by:  isperf2  22211  perflp  22213  perftop  22215  restperf  22243