Theorem ispcmp 34001

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 3450	. 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2		elex 3450	. 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3		id 22	. . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
4		fveq2 6840	. . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5		crefeq 33989	. . . . 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 2830	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8		df-pcmp 34000	. . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
9	7, 8	elab2g 3623	. 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 1542   ∈ wcel 2114  Vcvv 3429  ‘cfv 6498  LocFinclocfin 23469  CovHasRefccref 33986  Paracompcpcmp 33999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-cref 33987  df-pcmp 34000

This theorem is referenced by:  cmppcmp  34002  dispcmp  34003  pcmplfin  34004