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Theorem ispcmp 31151
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 6659 . . . . 5 (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5 crefeq 31139 . . . . 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 2910 . . 3 (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8 df-pcmp 31150 . . 3 Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}
97, 8elab2g 3654 . 2 (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
101, 2, 9pm5.21nii 383 1 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2115  Vcvv 3480  cfv 6344  LocFinclocfin 22107  CovHasRefccref 31136  Paracompcpcmp 31149
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352  df-cref 31137  df-pcmp 31150
This theorem is referenced by:  cmppcmp  31152  dispcmp  31153  pcmplfin  31154
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