Theorem ispcmp 33831

Description: The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
ispcmp	⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))

Proof of Theorem ispcmp

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2		elex 3484	. 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3		id 22	. . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
4		fveq2 6886	. . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5		crefeq 33819	. . . . 5 ⊢ ((LocFin‘𝑗) = (LocFin‘𝐽) → CovHasRef(LocFin‘𝑗) = CovHasRef(LocFin‘𝐽))
6	4, 5	syl 17	. . . 4 ⊢ (𝑗 = 𝐽 → CovHasRef(LocFin‘𝑗) = CovHasRef(LocFin‘𝐽))
7	3, 6	eleq12d 2827	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8		df-pcmp 33830	. . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
9	7, 8	elab2g 3663	. 2 ⊢ (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
10	1, 2, 9	pm5.21nii 378	1 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ‘cfv 6541  LocFinclocfin 23459  CovHasRefccref 33816  Paracompcpcmp 33829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-cref 33817  df-pcmp 33830

This theorem is referenced by:  cmppcmp  33832  dispcmp  33833  pcmplfin  33834