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Theorem ispcmp 33834
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.
StepHypRef Expression
1 elex 3500 . 2 (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2 elex 3500 . 2 (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3 id 22 . . . 4 (𝑗 = 𝐽𝑗 = 𝐽)
4 fveq2 6904 . . . . 5 (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5 crefeq 33822 . . . . 5 ((LocFin‘𝑗) = (LocFin‘𝐽) → CovHasRef(LocFin‘𝑗) = CovHasRef(LocFin‘𝐽))
64, 5syl 17 . . . 4 (𝑗 = 𝐽 → CovHasRef(LocFin‘𝑗) = CovHasRef(LocFin‘𝐽))
73, 6eleq12d 2834 . . 3 (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8 df-pcmp 33833 . . 3 Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}
97, 8elab2g 3679 . 2 (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
101, 2, 9pm5.21nii 378 1 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  Vcvv 3479  cfv 6559  LocFinclocfin 23502  CovHasRefccref 33819  Paracompcpcmp 33832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-cref 33820  df-pcmp 33833
This theorem is referenced by:  cmppcmp  33835  dispcmp  33836  pcmplfin  33837
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