![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 3413 | . 2 ⊢ (𝐽 ∈ Paracomp → 𝐽 ∈ V) | |
2 | elex 3413 | . 2 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) → 𝐽 ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽) | |
4 | fveq2 6446 | . . . . 5 ⊢ (𝑗 = 𝐽 → (LocFin‘𝑗) = (LocFin‘𝐽)) | |
5 | crefeq 30510 | . . . . 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 2852 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ CovHasRef(LocFin‘𝑗) ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
8 | df-pcmp 30521 | . . 3 ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} | |
9 | 7, 8 | elab2g 3560 | . 2 ⊢ (𝐽 ∈ V → (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))) |
10 | 1, 2, 9 | pm5.21nii 370 | 1 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ‘cfv 6135 LocFinclocfin 21716 CovHasRefccref 30507 Paracompcpcmp 30520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-cref 30508 df-pcmp 30521 |
This theorem is referenced by: cmppcmp 30523 dispcmp 30524 pcmplfin 30525 |
Copyright terms: Public domain | W3C validator |