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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 22892 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | unitg 22890 | . . 3 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵) | |
3 | 2 | eqcomd 2734 | . 2 ⊢ (𝐵 ∈ TopBases → ∪ 𝐵 = ∪ (topGen‘𝐵)) |
4 | istopon 22834 | . 2 ⊢ ((topGen‘𝐵) ∈ (TopOn‘∪ 𝐵) ↔ ((topGen‘𝐵) ∈ Top ∧ ∪ 𝐵 = ∪ (topGen‘𝐵))) | |
5 | 1, 3, 4 | sylanbrc 581 | 1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cuni 4912 ‘cfv 6553 topGenctg 17426 Topctop 22815 TopOnctopon 22832 TopBasesctb 22868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-topgen 17432 df-top 22816 df-topon 22833 df-bases 22869 |
This theorem is referenced by: ordttopon 23117 tgqtop 23636 alexsublem 23968 alexsub 23969 mopntopon 24365 topjoin 35882 istoprelowl 36872 |
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