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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 3459 | . 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V) | |
2 | elex 3459 | . 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽) | |
4 | fveq2 6645 | . . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽)) | |
5 | crefeq 31198 | . . . . 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 2884 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
8 | df-pcmp 31209 | . . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} | |
9 | 7, 8 | elab2g 3616 | . 2 ⊢ (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
10 | 1, 2, 9 | pm5.21nii 383 | 1 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 LocFinclocfin 22109 CovHasRefccref 31195 Paracompcpcmp 31208 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
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 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-cref 31196 df-pcmp 31209 |
This theorem is referenced by: cmppcmp 31211 dispcmp 31212 pcmplfin 31213 |
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