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Theorem ispcmp 33781
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 3498 . 2 (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2 elex 3498 . 2 (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3 id 22 . . . 4 (𝑗 = 𝐽𝑗 = 𝐽)
4 fveq2 6901 . . . . 5 (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5 crefeq 33769 . . . . 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 2831 . . 3 (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8 df-pcmp 33780 . . 3 Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}
97, 8elab2g 3683 . 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 1535  wcel 2104  Vcvv 3477  cfv 6558  LocFinclocfin 23509  CovHasRefccref 33766  Paracompcpcmp 33779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-iota 6510  df-fv 6566  df-cref 33767  df-pcmp 33780
This theorem is referenced by:  cmppcmp  33782  dispcmp  33783  pcmplfin  33784
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