Theorem isperf 23095

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 6834	. . . 4 ⊢ (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2		unieq 4874	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		lpfval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2789	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
5	1, 4	fveq12d 6841	. . 3 ⊢ (𝑗 = 𝐽 → ((limPt‘𝑗)‘∪ 𝑗) = ((limPt‘𝐽)‘𝑋))
6	5, 4	eqeq12d 2752	. 2 ⊢ (𝑗 = 𝐽 → (((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7		df-perf 23081	. 2 ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
8	6, 7	elrab2 3649	1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∪ cuni 4863  ‘cfv 6492  Topctop 22837  limPtclp 23078  Perfcperf 23079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-perf 23081

This theorem is referenced by:  isperf2  23096  perflp  23098  perftop  23100  restperf  23128