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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 21823 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
4 | 3 | simp2bi 1126 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2964 ∀wral 3085 ∃wrex 3086 {crab 3089 ∩ cin 3827 ∅c0 4177 ∪ cuni 4710 ‘cfv 6186 Fincfn 8302 Topctop 21199 LocFinclocfin 21810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ral 3090 df-rex 3091 df-rab 3094 df-v 3414 df-sbc 3681 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-op 4446 df-uni 4711 df-br 4928 df-opab 4990 df-mpt 5007 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fv 6194 df-top 21200 df-locfin 21813 |
This theorem is referenced by: lfinpfin 21830 lfinun 21831 locfincmp 21832 locfindis 21836 locfincf 21837 |
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