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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 21519 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 21518 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2798 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 21511 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2890 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 TopOnctopon 21515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-topon 21516 |
This theorem is referenced by: topgele 21535 eltpsg 21548 en2top 21590 resttopon 21766 ordtrest 21807 ordtrest2lem 21808 ordtrest2 21809 lmfval 21837 cnpfval 21839 iscn 21840 iscnp 21842 lmbrf 21865 cncls 21879 cnconst2 21888 cnrest2 21891 cndis 21896 cnindis 21897 cnpdis 21898 lmfss 21901 lmres 21905 lmff 21906 ist1-3 21954 connsuba 22025 unconn 22034 kgenval 22140 elkgen 22141 kgentopon 22143 pttoponconst 22202 tx1cn 22214 tx2cn 22215 ptcls 22221 xkoccn 22224 txlm 22253 cnmpt2res 22282 xkoinjcn 22292 qtoprest 22322 ordthmeolem 22406 pt1hmeo 22411 xkocnv 22419 flimclslem 22589 flfval 22595 flfnei 22596 isflf 22598 flfcnp 22609 txflf 22611 supnfcls 22625 fclscf 22630 fclscmp 22635 fcfval 22638 isfcf 22639 uffcfflf 22644 cnpfcf 22646 mopnm 23051 isxms2 23055 prdsxmslem2 23136 bcth2 23934 dvmptid 24560 dvmptc 24561 dvtaylp 24965 taylthlem1 24968 taylthlem2 24969 pige3ALT 25112 dvcxp1 25329 cxpcn3 25337 ordtrestNEW 31274 ordtrest2NEWlem 31275 ordtrest2NEW 31276 topjoin 33826 areacirclem1 35145 |
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