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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 20717 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 ‘cfv 5888 Topctop 20698 TopOnctopon 20715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topon 20716 |
This theorem is referenced by: topontopi 20720 topontopon 20724 toprntopon 20729 toponmax 20730 topgele 20734 istps 20738 en2top 20789 pptbas 20812 toponmre 20897 cldmreon 20898 iscldtop 20899 neiptopreu 20937 resttopon 20965 resttopon2 20972 restlp 20987 restperf 20988 perfopn 20989 ordtopn3 21000 ordtcld1 21001 ordtcld2 21002 ordttop 21004 lmfval 21036 cnfval 21037 cnpfval 21038 tgcn 21056 tgcnp 21057 subbascn 21058 iscnp4 21067 iscncl 21073 cncls2 21077 cncls 21078 cnntr 21079 cncnp 21084 cnindis 21096 lmcls 21106 iscnrm2 21142 ist0-2 21148 ist1-2 21151 ishaus2 21155 hausnei2 21157 isreg2 21181 sscmp 21208 dfconn2 21222 clsconn 21233 conncompcld 21237 1stccnp 21265 locfincf 21334 kgenval 21338 kgenftop 21343 1stckgenlem 21356 kgen2ss 21358 txtopon 21394 pttopon 21399 txcls 21407 ptclsg 21418 dfac14lem 21420 xkoccn 21422 txcnp 21423 ptcnplem 21424 txlm 21451 cnmpt2res 21480 cnmptkp 21483 cnmptk1 21484 cnmpt1k 21485 cnmptkk 21486 cnmptk1p 21488 cnmptk2 21489 xkoinjcn 21490 qtoptopon 21507 qtopcld 21516 qtoprest 21520 qtopcmap 21522 kqval 21529 regr1lem 21542 kqreglem1 21544 kqreglem2 21545 kqnrmlem1 21546 kqnrmlem2 21547 kqtop 21548 pt1hmeo 21609 xpstopnlem1 21612 xkohmeo 21618 neifil 21684 trnei 21696 elflim 21775 flimss1 21777 flimopn 21779 fbflim2 21781 flimcf 21786 flimclslem 21788 flffval 21793 flfnei 21795 flftg 21800 cnpflf2 21804 isfcls2 21817 fclsopn 21818 fclsnei 21823 fclscf 21829 fclscmp 21834 fcfval 21837 fcfnei 21839 cnpfcf 21845 tgpmulg2 21898 tmdgsum 21899 tmdgsum2 21900 subgntr 21910 opnsubg 21911 clssubg 21912 clsnsg 21913 cldsubg 21914 snclseqg 21919 tgphaus 21920 qustgpopn 21923 prdstgpd 21928 tsmsgsum 21942 tsmsid 21943 tgptsmscld 21954 mopntop 22245 metdseq0 22657 cnmpt2pc 22727 ishtpy 22771 om1val 22830 pi1val 22837 csscld 23048 clsocv 23049 relcmpcmet 23115 bcth2 23127 limcres 23650 perfdvf 23667 dvaddbr 23701 dvmulbr 23702 dvcmulf 23708 dvmptres2 23725 dvmptcmul 23727 dvmptntr 23734 dvcnvlem 23739 lhop2 23778 lhop 23779 dvcnvrelem2 23781 taylthlem1 24127 neibastop2 32356 neibastop3 32357 topjoin 32360 dissneqlem 33187 istopclsd 37263 dvresntr 40132 |
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