Theorem ispcmp 33367

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 3487	. 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2		elex 3487	. 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3		id 22	. . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
4		fveq2 6884	. . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5		crefeq 33355	. . . . 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 2821	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8		df-pcmp 33366	. . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
9	7, 8	elab2g 3665	. 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 205   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ‘cfv 6536  LocFinclocfin 23359  CovHasRefccref 33352  Paracompcpcmp 33365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-cref 33353  df-pcmp 33366

This theorem is referenced by:  cmppcmp  33368  dispcmp  33369  pcmplfin  33370