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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 23167 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) = ((topGen‘𝐽) ↾t 𝐴)) | |
| 2 | tgtop 22980 | . . . . 5 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘𝐽) = 𝐽) |
| 4 | 3 | oveq1d 7446 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((topGen‘𝐽) ↾t 𝐴) = (𝐽 ↾t 𝐴)) |
| 5 | 1, 4 | eqtrd 2777 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) = (𝐽 ↾t 𝐴)) |
| 6 | topbas 22979 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases) | |
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ TopBases) |
| 8 | restbas 23166 | . . 3 ⊢ (𝐽 ∈ TopBases → (𝐽 ↾t 𝐴) ∈ TopBases) | |
| 9 | tgcl 22976 | . . 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 2842 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 topGenctg 17482 Topctop 22899 TopBasesctb 22952 |
| 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-rep 5279 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-3or 1088 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-bases 22953 |
| This theorem is referenced by: resttopon 23169 resttopon2 23176 rest0 23177 restcld 23180 neitr 23188 restcls 23189 restntr 23190 ordtrest 23210 cmpsub 23408 fiuncmp 23412 1stcrest 23461 subislly 23489 llyrest 23493 nllyrest 23494 toplly 23498 cldllycmp 23503 kgencmp2 23554 llycmpkgen2 23558 1stckgen 23562 txkgen 23660 cnextfres1 24076 zdis 24838 cnmpopc 24955 dvbss 25936 dvreslem 25944 dvres2lem 25945 dvcnp2 25955 dvcnp2OLD 25956 dvmptres 26001 ulmdvlem3 26445 psercn 26470 abelth 26485 zarmxt1 33879 ordtrestNEW 33920 cvxpconn 35247 cvmscld 35278 ptrest 37626 poimirlem29 37656 cnambfre 37675 limcresiooub 45657 limcresioolb 45658 cncfuni 45901 cncfiooicclem1 45908 fourierdlem32 46154 fourierdlem33 46155 fourierdlem48 46169 fourierdlem49 46170 fouriersw 46246 iscnrm3lem1 48831 iscnrm3rlem7 48843 |
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