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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 2823 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2823 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
3 | 1, 2 | islocfin 22127 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐴 ∧ ∀𝑠 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
4 | 3 | simp1bi 1141 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 ∩ cin 3937 ∅c0 4293 ∪ cuni 4840 ‘cfv 6357 Fincfn 8511 Topctop 21503 LocFinclocfin 22114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-top 21504 df-locfin 22117 |
This theorem is referenced by: lfinun 22135 locfinreflem 31106 locfinref 31107 |
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