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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 21761 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) = ((topGen‘𝐽) ↾t 𝐴)) | |
2 | tgtop 21575 | . . . . 5 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘𝐽) = 𝐽) |
4 | 3 | oveq1d 7165 | . . 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 21574 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ TopBases) |
8 | restbas 21760 | . . 3 ⊢ (𝐽 ∈ TopBases → (𝐽 ↾t 𝐴) ∈ TopBases) | |
9 | tgcl 21571 | . . 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 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 ↾t crest 16688 topGenctg 16705 Topctop 21495 TopBasesctb 21547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-rest 16690 df-topgen 16711 df-top 21496 df-bases 21548 |
This theorem is referenced by: resttopon 21763 resttopon2 21770 rest0 21771 restcld 21774 neitr 21782 restcls 21783 restntr 21784 ordtrest 21804 cmpsub 22002 fiuncmp 22006 1stcrest 22055 subislly 22083 llyrest 22087 nllyrest 22088 toplly 22092 cldllycmp 22097 kgencmp2 22148 llycmpkgen2 22152 1stckgen 22156 txkgen 22254 cnextfres1 22670 zdis 23418 cnmpopc 23526 dvbss 24493 dvreslem 24501 dvres2lem 24502 dvcnp2 24511 dvmptres 24554 ulmdvlem3 24984 psercn 25008 abelth 25023 ordtrestNEW 31159 cvxpconn 32484 cvmscld 32515 ptrest 34885 poimirlem29 34915 cnambfre 34934 limcresiooub 41915 limcresioolb 41916 cncfuni 42161 cncfiooicclem1 42168 fourierdlem32 42417 fourierdlem33 42418 fourierdlem48 42432 fourierdlem49 42433 fouriersw 42509 |
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