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Mirrors > Home > MPE Home > Th. List > toponmax | Structured version Visualization version GIF version |
Description: The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toponmax | ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 21450 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 21449 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2818 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 21442 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2910 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cuni 4830 ‘cfv 6348 Topctop 21429 TopOnctopon 21446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-top 21430 df-topon 21447 |
This theorem is referenced by: topgele 21466 eltpsg 21479 en2top 21521 resttopon 21697 ordtrest 21738 ordtrest2lem 21739 ordtrest2 21740 lmfval 21768 cnpfval 21770 iscn 21771 iscnp 21773 lmbrf 21796 cncls 21810 cnconst2 21819 cnrest2 21822 cndis 21827 cnindis 21828 cnpdis 21829 lmfss 21832 lmres 21836 lmff 21837 ist1-3 21885 connsuba 21956 unconn 21965 kgenval 22071 elkgen 22072 kgentopon 22074 pttoponconst 22133 tx1cn 22145 tx2cn 22146 ptcls 22152 xkoccn 22155 txlm 22184 cnmpt2res 22213 xkoinjcn 22223 qtoprest 22253 ordthmeolem 22337 pt1hmeo 22342 xkocnv 22350 flimclslem 22520 flfval 22526 flfnei 22527 isflf 22529 flfcnp 22540 txflf 22542 supnfcls 22556 fclscf 22561 fclscmp 22566 fcfval 22569 isfcf 22570 uffcfflf 22575 cnpfcf 22577 mopnm 22981 isxms2 22985 prdsxmslem2 23066 bcth2 23860 dvmptid 24481 dvmptc 24482 dvtaylp 24885 taylthlem1 24888 taylthlem2 24889 pige3ALT 25032 dvcxp1 25248 cxpcn3 25256 ordtrestNEW 31063 ordtrest2NEWlem 31064 ordtrest2NEW 31065 topjoin 33610 areacirclem1 34863 |
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