Theorem isperf 23449

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 6877	. . . 4 ⊢ (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽))
2		unieq 4878	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		lpfval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2814	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
5	1, 4	fveq12d 6884	. . 3 ⊢ (𝑗 = 𝐽 → ((limPt‘𝑗)‘∪ 𝑗) = ((limPt‘𝐽)‘𝑋))
6	5, 4	eqeq12d 2777	. 2 ⊢ (𝑗 = 𝐽 → (((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋))
7		df-perf 23435	. 2 ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
8	6, 7	elrab2 3649	1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∪ cuni 4867  ‘cfv 6531  Topctop 23191  limPtclp 23432  Perfcperf 23433

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-perf 23435

This theorem is used by:  isperf2  23450  perflp  23452  perftop  23454  restperf  23482