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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 21320 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
4 | 3 | simp2bi 1077 | 1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ∩ cin 3573 ∅c0 3915 ∪ cuni 4436 ‘cfv 5888 Fincfn 7955 Topctop 20698 LocFinclocfin 21307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-top 20699 df-locfin 21310 |
This theorem is referenced by: lfinpfin 21327 lfinun 21328 locfincmp 21329 locfindis 21333 locfincf 21334 |
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