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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 22119 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
4 | 3 | simp2bi 1142 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ∩ cin 3935 ∅c0 4291 ∪ cuni 4832 ‘cfv 6350 Fincfn 8503 Topctop 21495 LocFinclocfin 22106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fv 6358 df-top 21496 df-locfin 22109 |
This theorem is referenced by: lfinpfin 22126 lfinun 22127 locfincmp 22128 locfindis 22132 locfincf 22133 |
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