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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispcmp | Structured version Visualization version GIF version |
Description: The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
ispcmp | ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3498 | . 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V) | |
2 | elex 3498 | . 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽) | |
4 | fveq2 6901 | . . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽)) | |
5 | crefeq 33769 | . . . . 5 ⊢ ((LocFin‘𝑗) = (LocFin‘𝐽) → CovHasRef(LocFin‘𝑗) = CovHasRef(LocFin‘𝐽)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑗 = 𝐽 → CovHasRef(LocFin‘𝑗) = CovHasRef(LocFin‘𝐽)) |
7 | 3, 6 | eleq12d 2831 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
8 | df-pcmp 33780 | . . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} | |
9 | 7, 8 | elab2g 3683 | . 2 ⊢ (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
10 | 1, 2, 9 | pm5.21nii 378 | 1 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1535 ∈ wcel 2104 Vcvv 3477 ‘cfv 6558 LocFinclocfin 23509 CovHasRefccref 33766 Paracompcpcmp 33779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-iota 6510 df-fv 6566 df-cref 33767 df-pcmp 33780 |
This theorem is referenced by: cmppcmp 33782 dispcmp 33783 pcmplfin 33784 |
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