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Mirrors > Home > MPE Home > Th. List > lfinpfin | Structured version Visualization version GIF version |
Description: A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
lfinpfin | ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐴 ∈ PtFin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2819 | . . . . . . . 8 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
3 | 1, 2 | locfinbas 22122 | . . . . . . 7 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∪ 𝐽 = ∪ 𝐴) |
4 | 3 | eleq2d 2896 | . . . . . 6 ⊢ (𝐴 ∈ (LocFin‘𝐽) → (𝑥 ∈ ∪ 𝐽 ↔ 𝑥 ∈ ∪ 𝐴)) |
5 | 4 | biimpar 480 | . . . . 5 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) → 𝑥 ∈ ∪ 𝐽) |
6 | 1 | locfinnei 22123 | . . . . 5 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) |
7 | 5, 6 | syldan 593 | . . . 4 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) |
8 | inelcm 4412 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑥 ∈ 𝑛) → (𝑠 ∩ 𝑛) ≠ ∅) | |
9 | 8 | expcom 416 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑛 → (𝑥 ∈ 𝑠 → (𝑠 ∩ 𝑛) ≠ ∅)) |
10 | 9 | ad2antlr 725 | . . . . . . . 8 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑥 ∈ 𝑛) ∧ 𝑠 ∈ 𝐴) → (𝑥 ∈ 𝑠 → (𝑠 ∩ 𝑛) ≠ ∅)) |
11 | 10 | ss2rabdv 4050 | . . . . . . 7 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑥 ∈ 𝑛) → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ⊆ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) |
12 | ssfi 8730 | . . . . . . . 8 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ⊆ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin) | |
13 | 12 | expcom 416 | . . . . . . 7 ⊢ ({𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ⊆ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin)) |
14 | 11, 13 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑥 ∈ 𝑛) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin)) |
15 | 14 | expimpd 456 | . . . . 5 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin)) |
16 | 15 | rexlimdvw 3288 | . . . 4 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin)) |
17 | 7, 16 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐴) → {𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin) |
18 | 17 | ralrimiva 3180 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ ∪ 𝐴{𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin) |
19 | 2 | isptfin 22116 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ ∪ 𝐴{𝑠 ∈ 𝐴 ∣ 𝑥 ∈ 𝑠} ∈ Fin)) |
20 | 18, 19 | mpbird 259 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐴 ∈ PtFin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ∃wrex 3137 {crab 3140 ∩ cin 3933 ⊆ wss 3934 ∅c0 4289 ∪ cuni 4830 ‘cfv 6348 Fincfn 8501 PtFincptfin 22103 LocFinclocfin 22104 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7573 df-er 8281 df-en 8502 df-fin 8505 df-top 21494 df-ptfin 22106 df-locfin 22107 |
This theorem is referenced by: locfindis 22130 |
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