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Mirrors > Home > MPE Home > Th. List > Mathboxes > pcmplfinf | Structured version Visualization version GIF version |
Description: Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
Ref | Expression |
---|---|
pcmplfin.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
pcmplfinf | ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcmplfin.x | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | simpll2 1215 | . . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑈 ⊆ 𝐽) | |
3 | simpll3 1216 | . . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑋 = ∪ 𝑈) | |
4 | elpwi 4512 | . . . 4 ⊢ (𝑣 ∈ 𝒫 𝐽 → 𝑣 ⊆ 𝐽) | |
5 | 4 | ad2antlr 727 | . . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ⊆ 𝐽) |
6 | simprr 773 | . . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣Ref𝑈) | |
7 | simprl 771 | . . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ∈ (LocFin‘𝐽)) | |
8 | 1, 2, 3, 5, 6, 7 | locfinref 31477 | . 2 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) |
9 | 1 | pcmplfin 31496 | . 2 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) |
10 | 8, 9 | r19.29a 3201 | 1 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ⊆ wss 3857 𝒫 cpw 4503 ∪ cuni 4809 class class class wbr 5043 ran crn 5541 ⟶wf 6365 ‘cfv 6369 Refcref 22371 LocFinclocfin 22373 Paracompcpcmp 31491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-reg 9197 ax-inf2 9245 ax-ac2 10060 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-fin 8619 df-r1 9363 df-rank 9364 df-card 9538 df-ac 9713 df-top 21763 df-topon 21780 df-ref 22374 df-locfin 22376 df-cref 31479 df-pcmp 31492 |
This theorem is referenced by: (None) |
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