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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 21151 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
4 | 3 | simprbi 492 | 1 ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∪ cuni 4673 ‘cfv 6137 Basecbs 16259 TopOpenctopn 16472 Topctop 21109 TopSpctps 21148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-top 21110 df-topon 21127 df-topsp 21149 |
This theorem is referenced by: mreclatdemoBAD 21312 haustsms 22351 cnextucn 22519 ressxms 22742 rlmbn 23571 rrhf 30644 esumcocn 30744 sibf0 30998 sibfof 31004 sitgclg 31006 sitgaddlemb 31012 sitmcl 31015 binomcxplemdvbinom 39518 binomcxplemnotnn0 39521 qndenserrn 41453 |
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