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Mirrors > Home > MPE Home > Th. List > tgtopon | Structured version Visualization version GIF version |
Description: A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
tgtopon | ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcl 22792 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | unitg 22790 | . . 3 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵) | |
3 | 2 | eqcomd 2737 | . 2 ⊢ (𝐵 ∈ TopBases → ∪ 𝐵 = ∪ (topGen‘𝐵)) |
4 | istopon 22734 | . 2 ⊢ ((topGen‘𝐵) ∈ (TopOn‘∪ 𝐵) ↔ ((topGen‘𝐵) ∈ Top ∧ ∪ 𝐵 = ∪ (topGen‘𝐵))) | |
5 | 1, 3, 4 | sylanbrc 582 | 1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∪ cuni 4908 ‘cfv 6543 topGenctg 17390 Topctop 22715 TopOnctopon 22732 TopBasesctb 22768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-topgen 17396 df-top 22716 df-topon 22733 df-bases 22769 |
This theorem is referenced by: ordttopon 23017 tgqtop 23536 alexsublem 23868 alexsub 23869 mopntopon 24265 topjoin 35714 istoprelowl 36705 |
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