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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 20542 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 20541 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2610 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 20536 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2688 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 TopOnctopon 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-topon 20523 |
This theorem is referenced by: topgele 20549 eltpsg 20560 en2top 20600 resttopon 20775 ordtrest 20816 ordtrest2lem 20817 ordtrest2 20818 lmfval 20846 cnpfval 20848 iscn 20849 iscnp 20851 lmbrf 20874 cncls 20888 cnconst2 20897 cnrest2 20900 cndis 20905 cnindis 20906 cnpdis 20907 lmfss 20910 lmres 20914 lmff 20915 ist1-3 20963 consuba 21033 uncon 21042 kgenval 21148 elkgen 21149 kgentopon 21151 pttoponconst 21210 tx1cn 21222 tx2cn 21223 ptcls 21229 xkoccn 21232 txlm 21261 cnmpt2res 21290 xkoinjcn 21300 qtoprest 21330 ordthmeolem 21414 pt1hmeo 21419 xkocnv 21427 flimclslem 21598 flfval 21604 flfnei 21605 isflf 21607 flfcnp 21618 txflf 21620 supnfcls 21634 fclscf 21639 fclscmp 21644 fcfval 21647 isfcf 21648 uffcfflf 21653 cnpfcf 21655 mopnm 22059 isxms2 22063 prdsxmslem2 22144 bcth2 22935 dvmptid 23526 dvmptc 23527 dvtaylp 23928 taylthlem1 23931 taylthlem2 23932 pige3 24073 dvcxp1 24281 cxpcn3 24289 ordtrestNEW 29295 ordtrest2NEWlem 29296 ordtrest2NEW 29297 topjoin 31530 areacirclem1 32670 |
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