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Theorem ispcmp 29909
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 3210 . 2 (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2 elex 3210 . 2 (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3 id 22 . . . 4 (𝑗 = 𝐽𝑗 = 𝐽)
4 fveq2 6189 . . . . 5 (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5 crefeq 29897 . . . . 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 2694 . . 3 (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8 df-pcmp 29908 . . 3 Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}
97, 8elab2g 3351 . 2 (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
101, 2, 9pm5.21nii 368 1 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1482  wcel 1989  Vcvv 3198  cfv 5886  LocFinclocfin 21301  CovHasRefccref 29894  Paracompcpcmp 29907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-cref 29895  df-pcmp 29908
This theorem is referenced by:  cmppcmp  29910  dispcmp  29911  pcmplfin  29912
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