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Theorem ispcmp 33463
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 3490 . 2 (𝐽 ∈ Paracomp β†’ 𝐽 ∈ V)
2 elex 3490 . 2 (𝐽 ∈ CovHasRef(LocFinβ€˜π½) β†’ 𝐽 ∈ V)
3 id 22 . . . 4 (𝑗 = 𝐽 β†’ 𝑗 = 𝐽)
4 fveq2 6900 . . . . 5 (𝑗 = 𝐽 β†’ (LocFinβ€˜π‘—) = (LocFinβ€˜π½))
5 crefeq 33451 . . . . 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 2822 . . 3 (𝑗 = 𝐽 β†’ (𝑗 ∈ CovHasRef(LocFinβ€˜π‘—) ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½)))
8 df-pcmp 33462 . . 3 Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFinβ€˜π‘—)}
97, 8elab2g 3669 . 2 (𝐽 ∈ V β†’ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½)))
101, 2, 9pm5.21nii 377 1 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3471  β€˜cfv 6551  LocFinclocfin 23426  CovHasRefccref 33448  Paracompcpcmp 33461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-cref 33449  df-pcmp 33462
This theorem is referenced by:  cmppcmp  33464  dispcmp  33465  pcmplfin  33466
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