Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 21786 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
4 | 3 | simprbi 500 | 1 ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∪ cuni 4805 ‘cfv 6358 Basecbs 16666 TopOpenctopn 16880 Topctop 21744 TopSpctps 21783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-top 21745 df-topon 21762 df-topsp 21784 |
This theorem is referenced by: mreclatdemoBAD 21947 haustsms 22987 cnextucn 23154 ressxms 23377 rlmbn 24212 rrhf 31614 esumcocn 31714 sibf0 31967 sibfof 31973 sitgclg 31975 sitgaddlemb 31981 sitmcl 31984 binomcxplemdvbinom 41585 binomcxplemnotnn0 41588 qndenserrn 43458 |
Copyright terms: Public domain | W3C validator |