Theorem ispcmp 33865

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 3457	. 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2		elex 3457	. 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3		id 22	. . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
4		fveq2 6822	. . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5		crefeq 33853	. . . . 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 2825	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8		df-pcmp 33864	. . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
9	7, 8	elab2g 3636	. 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 1541   ∈ wcel 2111  Vcvv 3436  ‘cfv 6481  LocFinclocfin 23417  CovHasRefccref 33850  Paracompcpcmp 33863

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-cref 33851  df-pcmp 33864

This theorem is referenced by:  cmppcmp  33866  dispcmp  33867  pcmplfin  33868