Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispcmp Structured version   Visualization version   GIF version

Theorem ispcmp 30522
 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 3413 . 2 (𝐽 ∈ Paracomp → 𝐽 ∈ V)
2 elex 3413 . 2 (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V)
3 id 22 . . . 4 (𝑗 = 𝐽𝑗 = 𝐽)
4 fveq2 6446 . . . . 5 (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽))
5 crefeq 30510 . . . . 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 2852 . . 3 (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
8 df-pcmp 30521 . . 3 Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}
97, 8elab2g 3560 . 2 (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)))
101, 2, 9pm5.21nii 370 1 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1601   ∈ wcel 2106  Vcvv 3397  ‘cfv 6135  LocFinclocfin 21716  CovHasRefccref 30507  Paracompcpcmp 30520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-cref 30508  df-pcmp 30521 This theorem is referenced by:  cmppcmp  30523  dispcmp  30524  pcmplfin  30525
 Copyright terms: Public domain W3C validator