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Mirrors > Home > MPE Home > Th. List > locfinbas | Structured version Visualization version GIF version |
Description: A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Ref | Expression |
---|---|
locfinbas.1 | ⊢ 𝑋 = ∪ 𝐽 |
locfinbas.2 | ⊢ 𝑌 = ∪ 𝐴 |
Ref | Expression |
---|---|
locfinbas | ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | locfinbas.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | locfinbas.2 | . . 3 ⊢ 𝑌 = ∪ 𝐴 | |
3 | 1, 2 | islocfin 21541 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
4 | 3 | simp2bi 1140 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 ∃wrex 3062 {crab 3065 ∩ cin 3722 ∅c0 4063 ∪ cuni 4575 ‘cfv 6030 Fincfn 8113 Topctop 20918 LocFinclocfin 21528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fv 6038 df-top 20919 df-locfin 21531 |
This theorem is referenced by: lfinpfin 21548 lfinun 21549 locfincmp 21550 locfindis 21554 locfincf 21555 |
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