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Mirrors > Home > MPE Home > Th. List > locfintop | Structured version Visualization version GIF version |
Description: A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Ref | Expression |
---|---|
locfintop | ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2818 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
3 | 1, 2 | islocfin 22053 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐴 ∧ ∀𝑠 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
4 | 3 | simp1bi 1137 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 {crab 3139 ∩ cin 3932 ∅c0 4288 ∪ cuni 4830 ‘cfv 6348 Fincfn 8497 Topctop 21429 LocFinclocfin 22040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-iota 6307 df-fun 6350 df-fv 6356 df-top 21430 df-locfin 22043 |
This theorem is referenced by: lfinun 22061 locfinreflem 31003 locfinref 31004 |
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