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Mirrors > Home > MPE Home > Th. List > tpsuni | Structured version Visualization version GIF version |
Description: The base set of a topological space. (Contributed by FL, 27-Jun-2014.) |
Ref | Expression |
---|---|
istps.a | ⊢ 𝐴 = (Base‘𝐾) |
istps.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
Ref | Expression |
---|---|
tpsuni | ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istps.a | . . 3 ⊢ 𝐴 = (Base‘𝐾) | |
2 | istps.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | istps2 22831 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
4 | 3 | simprbi 496 | 1 ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cuni 4904 ‘cfv 6543 Basecbs 17174 TopOpenctopn 17397 Topctop 22789 TopSpctps 22828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-top 22790 df-topon 22807 df-topsp 22829 |
This theorem is referenced by: mreclatdemoBAD 22994 haustsms 24034 cnextucn 24202 ressxms 24428 rlmbn 25283 rrhf 33594 esumcocn 33694 sibf0 33949 sibfof 33955 sitgclg 33957 sitgaddlemb 33963 sitmcl 33966 binomcxplemdvbinom 43781 binomcxplemnotnn0 43784 qndenserrn 45678 |
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