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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 2727 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | toptopon 22806 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2099 ∪ cuni 4903 ‘cfv 6542 Topctop 22782 TopOnctopon 22799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-topon 22800 |
This theorem is referenced by: topontopon 22808 toprntopon 22814 neiptopreu 23024 lmcvg 23153 cnss1 23167 cnss2 23168 cnrest2 23177 cnrest2r 23178 lmss 23189 lmcnp 23195 lmcn 23196 t1t0 23239 haust1 23243 restcnrm 23253 resthauslem 23254 lmmo 23271 rncmp 23287 connima 23316 conncn 23317 kgeni 23428 kgenftop 23431 kgenss 23434 kgenhaus 23435 kgencmp2 23437 kgenidm 23438 1stckgen 23445 kgencn3 23449 kgen2cn 23450 dfac14 23509 ptcnplem 23512 ptcnp 23513 txcnmpt 23515 ptcn 23518 txdis1cn 23526 lmcn2 23540 txkgen 23543 xkohaus 23544 xkopt 23546 cnmpt11 23554 cnmpt11f 23555 cnmpt1t 23556 cnmpt12 23558 cnmpt21 23562 cnmpt21f 23563 cnmpt2t 23564 cnmpt22 23565 cnmpt22f 23566 cnmptcom 23569 cnmptkp 23571 cnmpt2k 23579 txconn 23580 qtopss 23606 qtopeu 23607 qtopomap 23609 qtopcmap 23610 kqtop 23636 kqt0 23637 nrmr0reg 23640 regr1 23641 kqreg 23642 kqnrm 23643 hmeoqtop 23666 hmphref 23672 xpstopnlem1 23700 ptcmpfi 23704 xkocnv 23705 xkohmeo 23706 kqhmph 23710 flimsncls 23877 cnpflfi 23890 flfcnp 23895 flfcnp2 23898 cnpfcfi 23931 cnextucn 24195 cnmpopc 24836 htpyco1 24891 htpyco2 24892 phtpyco2 24903 pcopt 24936 pcopt2 24937 pcorevlem 24940 pi1cof 24973 pi1coghm 24975 cvxsconn 34789 clduni 47842 |
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