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Mirrors > Home > MPE Home > Th. List > resttop | Structured version Visualization version GIF version |
Description: A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. π΄ is normally a subset of the base set of π½. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
resttop | β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgrest 22663 | . . 3 β’ ((π½ β Top β§ π΄ β π) β (topGenβ(π½ βΎt π΄)) = ((topGenβπ½) βΎt π΄)) | |
2 | tgtop 22476 | . . . . 5 β’ (π½ β Top β (topGenβπ½) = π½) | |
3 | 2 | adantr 482 | . . . 4 β’ ((π½ β Top β§ π΄ β π) β (topGenβπ½) = π½) |
4 | 3 | oveq1d 7424 | . . 3 β’ ((π½ β Top β§ π΄ β π) β ((topGenβπ½) βΎt π΄) = (π½ βΎt π΄)) |
5 | 1, 4 | eqtrd 2773 | . 2 β’ ((π½ β Top β§ π΄ β π) β (topGenβ(π½ βΎt π΄)) = (π½ βΎt π΄)) |
6 | topbas 22475 | . . . 4 β’ (π½ β Top β π½ β TopBases) | |
7 | 6 | adantr 482 | . . 3 β’ ((π½ β Top β§ π΄ β π) β π½ β TopBases) |
8 | restbas 22662 | . . 3 β’ (π½ β TopBases β (π½ βΎt π΄) β TopBases) | |
9 | tgcl 22472 | . . 3 β’ ((π½ βΎt π΄) β TopBases β (topGenβ(π½ βΎt π΄)) β Top) | |
10 | 7, 8, 9 | 3syl 18 | . 2 β’ ((π½ β Top β§ π΄ β π) β (topGenβ(π½ βΎt π΄)) β Top) |
11 | 5, 10 | eqeltrrd 2835 | 1 β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 βΎt crest 17366 topGenctg 17383 Topctop 22395 TopBasesctb 22448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-en 8940 df-fin 8943 df-fi 9406 df-rest 17368 df-topgen 17389 df-top 22396 df-bases 22449 |
This theorem is referenced by: resttopon 22665 resttopon2 22672 rest0 22673 restcld 22676 neitr 22684 restcls 22685 restntr 22686 ordtrest 22706 cmpsub 22904 fiuncmp 22908 1stcrest 22957 subislly 22985 llyrest 22989 nllyrest 22990 toplly 22994 cldllycmp 22999 kgencmp2 23050 llycmpkgen2 23054 1stckgen 23058 txkgen 23156 cnextfres1 23572 zdis 24332 cnmpopc 24444 dvbss 25418 dvreslem 25426 dvres2lem 25427 dvcnp2 25437 dvmptres 25480 ulmdvlem3 25914 psercn 25938 abelth 25953 zarmxt1 32860 ordtrestNEW 32901 cvxpconn 34233 cvmscld 34264 gg-dvcnp2 35174 ptrest 36487 poimirlem29 36517 cnambfre 36536 limcresiooub 44358 limcresioolb 44359 cncfuni 44602 cncfiooicclem1 44609 fourierdlem32 44855 fourierdlem33 44856 fourierdlem48 44870 fourierdlem49 44871 fouriersw 44947 iscnrm3lem1 47566 iscnrm3rlem7 47579 |
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