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Mirrors > Home > ILE Home > Th. List > tgtopon | 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 12243 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | unitg 12241 | . . 3 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵) | |
3 | 2 | eqcomd 2145 | . 2 ⊢ (𝐵 ∈ TopBases → ∪ 𝐵 = ∪ (topGen‘𝐵)) |
4 | istopon 12190 | . 2 ⊢ ((topGen‘𝐵) ∈ (TopOn‘∪ 𝐵) ↔ ((topGen‘𝐵) ∈ Top ∧ ∪ 𝐵 = ∪ (topGen‘𝐵))) | |
5 | 1, 3, 4 | sylanbrc 413 | 1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∪ cuni 3736 ‘cfv 5123 topGenctg 12145 Topctop 12174 TopOnctopon 12187 TopBasesctb 12219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-topgen 12151 df-top 12175 df-topon 12188 df-bases 12220 |
This theorem is referenced by: mopntopon 12622 |
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