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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 22335 | . 2 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
2 | unitg 22333 | . . 3 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵) | |
3 | 2 | eqcomd 2739 | . 2 ⊢ (𝐵 ∈ TopBases → ∪ 𝐵 = ∪ (topGen‘𝐵)) |
4 | istopon 22277 | . 2 ⊢ ((topGen‘𝐵) ∈ (TopOn‘∪ 𝐵) ↔ ((topGen‘𝐵) ∈ Top ∧ ∪ 𝐵 = ∪ (topGen‘𝐵))) | |
5 | 1, 3, 4 | sylanbrc 584 | 1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cuni 4866 ‘cfv 6497 topGenctg 17324 Topctop 22258 TopOnctopon 22275 TopBasesctb 22311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-topgen 17330 df-top 22259 df-topon 22276 df-bases 22312 |
This theorem is referenced by: ordttopon 22560 tgqtop 23079 alexsublem 23411 alexsub 23412 mopntopon 23808 topjoin 34883 istoprelowl 35877 |
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