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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 21452 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 21451 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2821 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 21444 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2913 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cuni 4832 ‘cfv 6349 Topctop 21431 TopOnctopon 21448 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-top 21432 df-topon 21449 |
This theorem is referenced by: topgele 21468 eltpsg 21481 en2top 21523 resttopon 21699 ordtrest 21740 ordtrest2lem 21741 ordtrest2 21742 lmfval 21770 cnpfval 21772 iscn 21773 iscnp 21775 lmbrf 21798 cncls 21812 cnconst2 21821 cnrest2 21824 cndis 21829 cnindis 21830 cnpdis 21831 lmfss 21834 lmres 21838 lmff 21839 ist1-3 21887 connsuba 21958 unconn 21967 kgenval 22073 elkgen 22074 kgentopon 22076 pttoponconst 22135 tx1cn 22147 tx2cn 22148 ptcls 22154 xkoccn 22157 txlm 22186 cnmpt2res 22215 xkoinjcn 22225 qtoprest 22255 ordthmeolem 22339 pt1hmeo 22344 xkocnv 22352 flimclslem 22522 flfval 22528 flfnei 22529 isflf 22531 flfcnp 22542 txflf 22544 supnfcls 22558 fclscf 22563 fclscmp 22568 fcfval 22571 isfcf 22572 uffcfflf 22577 cnpfcf 22579 mopnm 22983 isxms2 22987 prdsxmslem2 23068 bcth2 23862 dvmptid 24483 dvmptc 24484 dvtaylp 24887 taylthlem1 24890 taylthlem2 24891 pige3ALT 25034 dvcxp1 25248 cxpcn3 25256 ordtrestNEW 31064 ordtrest2NEWlem 31065 ordtrest2NEW 31066 topjoin 33611 areacirclem1 34864 |
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