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 3212 | . 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V) | |
2 | elex 3212 | . 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽) | |
4 | fveq2 6191 | . . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽)) | |
5 | crefeq 29912 | . . . . 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 2695 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
8 | df-pcmp 29923 | . . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} | |
9 | 7, 8 | elab2g 3353 | . 2 ⊢ (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
10 | 1, 2, 9 | pm5.21nii 368 | 1 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ‘cfv 5888 LocFinclocfin 21307 CovHasRefccref 29909 Paracompcpcmp 29922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-cref 29910 df-pcmp 29923 |
This theorem is referenced by: cmppcmp 29925 dispcmp 29926 pcmplfin 29927 |
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