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Mirrors > Home > MPE Home > Th. List > toptopon | Structured version Visualization version GIF version |
Description: Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toptopon.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
toptopon | ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | istopon 20919 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
3 | 1, 2 | mpbiran2 992 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ Top) |
4 | 3 | bicomi 214 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1632 ∈ wcel 2139 ∪ cuni 4588 ‘cfv 6049 Topctop 20900 TopOnctopon 20917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-topon 20918 |
This theorem is referenced by: toptopon2 20925 eltpsi 20950 neiptopreu 21139 restuni 21168 stoig 21169 restlp 21189 restperf 21190 perfopn 21191 iscn2 21244 iscnp2 21245 lmcvg 21268 cnss1 21282 cnss2 21283 cncnpi 21284 cncnp2 21287 cnnei 21288 cnrest 21291 cnrest2 21292 cnrest2r 21293 cnpresti 21294 cnprest 21295 cnprest2 21296 paste 21300 lmss 21304 lmcnp 21310 lmcn 21311 t1t0 21354 haust1 21358 restcnrm 21368 resthauslem 21369 t1sep2 21375 sshauslem 21378 lmmo 21386 rncmp 21401 connima 21430 conncn 21431 1stcelcls 21466 kgeni 21542 kgenuni 21544 kgenftop 21545 kgenss 21548 kgenhaus 21549 kgencmp2 21551 kgenidm 21552 iskgen3 21554 1stckgen 21559 kgencn3 21563 kgen2cn 21564 txuni 21597 ptuniconst 21603 dfac14 21623 ptcnplem 21626 ptcnp 21627 txcnmpt 21629 txcn 21631 ptcn 21632 txindis 21639 txdis1cn 21640 ptrescn 21644 txcmpb 21649 lmcn2 21654 txkgen 21657 xkohaus 21658 xkoptsub 21659 xkopt 21660 cnmpt11 21668 cnmpt11f 21669 cnmpt1t 21670 cnmpt12 21672 cnmpt21 21676 cnmpt21f 21677 cnmpt2t 21678 cnmpt22 21679 cnmpt22f 21680 cnmptcom 21683 cnmptkp 21685 xkofvcn 21689 cnmpt2k 21693 txconn 21694 imasnopn 21695 imasncld 21696 imasncls 21697 qtopcmplem 21712 qtopkgen 21715 qtopss 21720 qtopeu 21721 qtopomap 21723 qtopcmap 21724 kqtop 21750 kqt0 21751 nrmr0reg 21754 regr1 21755 kqreg 21756 kqnrm 21757 hmeof1o 21769 hmeores 21776 hmeoqtop 21780 hmphref 21786 hmphindis 21802 cmphaushmeo 21805 txhmeo 21808 ptunhmeo 21813 xpstopnlem1 21814 ptcmpfi 21818 xkocnv 21819 xkohmeo 21820 kqhmph 21824 hausflim 21986 flimsncls 21991 flfneii 21997 hausflf 22002 cnpflfi 22004 flfcnp 22009 flfcnp2 22012 flimfnfcls 22033 cnpfcfi 22045 flfcntr 22048 cnextfun 22069 cnextfvval 22070 cnextf 22071 cnextcn 22072 cnextfres1 22073 cnextucn 22308 retopon 22768 cnmpt2pc 22928 evth 22959 evth2 22960 htpyco1 22978 htpyco2 22979 phtpyco2 22990 pcopt 23022 pcopt2 23023 pcorevlem 23026 pi1cof 23059 pi1coghm 23061 qtophaus 30212 rrhre 30374 pconnconn 31520 connpconn 31524 pconnpi1 31526 sconnpi1 31528 txsconnlem 31529 txsconn 31530 cvxsconn 31532 cvmsf1o 31561 cvmliftmolem1 31570 cvmliftlem8 31581 cvmlift2lem9a 31592 cvmlift2lem9 31600 cvmlift2lem11 31602 cvmlift2lem12 31603 cvmliftphtlem 31606 cvmlift3lem6 31613 cvmlift3lem8 31615 cvmlift3lem9 31616 cnres2 33875 cnresima 33876 hausgraph 38292 ntrf2 38924 fcnre 39683 |
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