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Theorem ispcmp 32250
Description: The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
ispcmp (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))

Proof of Theorem ispcmp
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 elex 3461 . 2 (𝐽 ∈ Paracomp β†’ 𝐽 ∈ V)
2 elex 3461 . 2 (𝐽 ∈ CovHasRef(LocFinβ€˜π½) β†’ 𝐽 ∈ V)
3 id 22 . . . 4 (𝑗 = 𝐽 β†’ 𝑗 = 𝐽)
4 fveq2 6839 . . . . 5 (𝑗 = 𝐽 β†’ (LocFinβ€˜π‘—) = (LocFinβ€˜π½))
5 crefeq 32238 . . . . 5 ((LocFinβ€˜π‘—) = (LocFinβ€˜π½) β†’ CovHasRef(LocFinβ€˜π‘—) = CovHasRef(LocFinβ€˜π½))
64, 5syl 17 . . . 4 (𝑗 = 𝐽 β†’ CovHasRef(LocFinβ€˜π‘—) = CovHasRef(LocFinβ€˜π½))
73, 6eleq12d 2832 . . 3 (𝑗 = 𝐽 β†’ (𝑗 ∈ CovHasRef(LocFinβ€˜π‘—) ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½)))
8 df-pcmp 32249 . . 3 Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFinβ€˜π‘—)}
97, 8elab2g 3630 . 2 (𝐽 ∈ V β†’ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½)))
101, 2, 9pm5.21nii 379 1 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3443  β€˜cfv 6493  LocFinclocfin 22807  CovHasRefccref 32235  Paracompcpcmp 32248
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-cref 32236  df-pcmp 32249
This theorem is referenced by:  cmppcmp  32251  dispcmp  32252  pcmplfin  32253
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