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Mirrors > Home > MPE Home > Th. List > topontop | Structured version Visualization version GIF version |
Description: A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
topontop | ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopon 21522 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cuni 4840 ‘cfv 6357 Topctop 21503 TopOnctopon 21520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-topon 21521 |
This theorem is referenced by: topontopi 21525 topontopon 21529 toprntopon 21535 toponmax 21536 topgele 21540 istps 21544 en2top 21595 pptbas 21618 toponmre 21703 cldmreon 21704 iscldtop 21705 neiptopreu 21743 resttopon 21771 resttopon2 21778 restlp 21793 restperf 21794 perfopn 21795 ordtopn3 21806 ordtcld1 21807 ordtcld2 21808 ordttop 21810 lmfval 21842 cnfval 21843 cnpfval 21844 tgcn 21862 tgcnp 21863 subbascn 21864 iscnp4 21873 iscncl 21879 cncls2 21883 cncls 21884 cnntr 21885 cncnp 21890 cnindis 21902 lmcls 21912 iscnrm2 21948 ist0-2 21954 ist1-2 21957 ishaus2 21961 hausnei2 21963 isreg2 21987 sscmp 22015 dfconn2 22029 clsconn 22040 conncompcld 22044 1stccnp 22072 locfincf 22141 kgenval 22145 kgenftop 22150 1stckgenlem 22163 kgen2ss 22165 txtopon 22201 pttopon 22206 txcls 22214 ptclsg 22225 dfac14lem 22227 xkoccn 22229 txcnp 22230 ptcnplem 22231 txlm 22258 cnmpt2res 22287 cnmptkp 22290 cnmptk1 22291 cnmpt1k 22292 cnmptkk 22293 cnmptk1p 22295 cnmptk2 22296 xkoinjcn 22297 qtoptopon 22314 qtopcld 22323 qtoprest 22327 qtopcmap 22329 kqval 22336 regr1lem 22349 kqreglem1 22351 kqreglem2 22352 kqnrmlem1 22353 kqnrmlem2 22354 kqtop 22355 pt1hmeo 22416 xpstopnlem1 22419 xkohmeo 22425 neifil 22490 trnei 22502 elflim 22581 flimss1 22583 flimopn 22585 fbflim2 22587 flimcf 22592 flimclslem 22594 flffval 22599 flfnei 22601 flftg 22606 cnpflf2 22610 isfcls2 22623 fclsopn 22624 fclsnei 22629 fclscf 22635 fclscmp 22640 fcfval 22643 fcfnei 22645 cnpfcf 22651 tgpmulg2 22704 tmdgsum 22705 tmdgsum2 22706 subgntr 22717 opnsubg 22718 clssubg 22719 clsnsg 22720 cldsubg 22721 snclseqg 22726 tgphaus 22727 qustgpopn 22730 prdstgpd 22735 tsmsgsum 22749 tsmsid 22750 tgptsmscld 22761 mopntop 23052 metdseq0 23464 cnmpopc 23534 ishtpy 23578 om1val 23636 pi1val 23643 csscld 23854 clsocv 23855 relcmpcmet 23923 bcth2 23935 limcres 24486 perfdvf 24503 dvaddbr 24537 dvmulbr 24538 dvcmulf 24544 dvmptres2 24561 dvmptcmul 24563 dvmptntr 24570 dvcnvlem 24575 lhop2 24614 lhop 24615 dvcnvrelem2 24617 taylthlem1 24963 neibastop2 33711 neibastop3 33712 topjoin 33715 dissneqlem 34623 istopclsd 39304 dvresntr 42209 |
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