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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. 𝐴 is normally a subset of the base set of 𝐽. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
resttop | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgrest 21697 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) = ((topGen‘𝐽) ↾t 𝐴)) | |
2 | tgtop 21511 | . . . . 5 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
3 | 2 | adantr 481 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘𝐽) = 𝐽) |
4 | 3 | oveq1d 7160 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((topGen‘𝐽) ↾t 𝐴) = (𝐽 ↾t 𝐴)) |
5 | 1, 4 | eqtrd 2856 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) = (𝐽 ↾t 𝐴)) |
6 | topbas 21510 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ TopBases) |
8 | restbas 21696 | . . 3 ⊢ (𝐽 ∈ TopBases → (𝐽 ↾t 𝐴) ∈ TopBases) | |
9 | tgcl 21507 | . . 3 ⊢ ((𝐽 ↾t 𝐴) ∈ TopBases → (topGen‘(𝐽 ↾t 𝐴)) ∈ Top) | |
10 | 7, 8, 9 | 3syl 18 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) ∈ Top) |
11 | 5, 10 | eqeltrrd 2914 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ↾t crest 16684 topGenctg 16701 Topctop 21431 TopBasesctb 21483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-oadd 8097 df-er 8279 df-en 8499 df-fin 8502 df-fi 8864 df-rest 16686 df-topgen 16707 df-top 21432 df-bases 21484 |
This theorem is referenced by: resttopon 21699 resttopon2 21706 rest0 21707 restcld 21710 neitr 21718 restcls 21719 restntr 21720 ordtrest 21740 cmpsub 21938 fiuncmp 21942 1stcrest 21991 subislly 22019 llyrest 22023 nllyrest 22024 toplly 22028 cldllycmp 22033 kgencmp2 22084 llycmpkgen2 22088 1stckgen 22092 txkgen 22190 cnextfres1 22606 zdis 23353 cnmpopc 23461 dvbss 24428 dvreslem 24436 dvres2lem 24437 dvcnp2 24446 dvmptres 24489 ulmdvlem3 24919 psercn 24943 abelth 24958 ordtrestNEW 31064 cvxpconn 32387 cvmscld 32418 ptrest 34773 poimirlem29 34803 cnambfre 34822 limcresiooub 41803 limcresioolb 41804 cncfuni 42049 cncfiooicclem1 42056 fourierdlem32 42305 fourierdlem33 42306 fourierdlem48 42320 fourierdlem49 42321 fouriersw 42397 |
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