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| Mirrors > Home > MPE Home > Th. List > toptopon2 | Structured version Visualization version GIF version | ||
| Description: A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
| Ref | Expression |
|---|---|
| toptopon2 | ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | toptopon 22923 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topon 22917 |
| This theorem is referenced by: topontopon 22925 toprntopon 22931 neiptopreu 23141 lmcvg 23270 cnss1 23284 cnss2 23285 cnrest2 23294 cnrest2r 23295 lmss 23306 lmcnp 23312 lmcn 23313 t1t0 23356 haust1 23360 restcnrm 23370 resthauslem 23371 lmmo 23388 rncmp 23404 connima 23433 conncn 23434 kgeni 23545 kgenftop 23548 kgenss 23551 kgenhaus 23552 kgencmp2 23554 kgenidm 23555 1stckgen 23562 kgencn3 23566 kgen2cn 23567 dfac14 23626 ptcnplem 23629 ptcnp 23630 txcnmpt 23632 ptcn 23635 txdis1cn 23643 lmcn2 23657 txkgen 23660 xkohaus 23661 xkopt 23663 cnmpt11 23671 cnmpt11f 23672 cnmpt1t 23673 cnmpt12 23675 cnmpt21 23679 cnmpt21f 23680 cnmpt2t 23681 cnmpt22 23682 cnmpt22f 23683 cnmptcom 23686 cnmptkp 23688 cnmpt2k 23696 txconn 23697 qtopss 23723 qtopeu 23724 qtopomap 23726 qtopcmap 23727 kqtop 23753 kqt0 23754 nrmr0reg 23757 regr1 23758 kqreg 23759 kqnrm 23760 hmeoqtop 23783 hmphref 23789 xpstopnlem1 23817 ptcmpfi 23821 xkocnv 23822 xkohmeo 23823 kqhmph 23827 flimsncls 23994 cnpflfi 24007 flfcnp 24012 flfcnp2 24015 cnpfcfi 24048 cnextucn 24312 cnmpopc 24955 htpyco1 25010 htpyco2 25011 phtpyco2 25022 pcopt 25055 pcopt2 25056 pcorevlem 25059 pi1cof 25092 pi1coghm 25094 cvxsconn 35248 clduni 48798 |
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