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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 13457 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | unitg 13455 | . . 3 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵) | |
3 | 2 | eqcomd 2183 | . 2 ⊢ (𝐵 ∈ TopBases → ∪ 𝐵 = ∪ (topGen‘𝐵)) |
4 | istopon 13404 | . 2 ⊢ ((topGen‘𝐵) ∈ (TopOn‘∪ 𝐵) ↔ ((topGen‘𝐵) ∈ Top ∧ ∪ 𝐵 = ∪ (topGen‘𝐵))) | |
5 | 1, 3, 4 | sylanbrc 417 | 1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∪ cuni 3809 ‘cfv 5216 topGenctg 12693 Topctop 13388 TopOnctopon 13401 TopBasesctb 13433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-topgen 12699 df-top 13389 df-topon 13402 df-bases 13434 |
This theorem is referenced by: mopntopon 13836 |
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