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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 21767 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(𝐽 ↾t 𝐴)) = ((topGen‘𝐽) ↾t 𝐴)) | |
2 | tgtop 21581 | . . . . 5 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘𝐽) = 𝐽) |
4 | 3 | oveq1d 7171 | . . 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 21580 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ TopBases) |
8 | restbas 21766 | . . 3 ⊢ (𝐽 ∈ TopBases → (𝐽 ↾t 𝐴) ∈ TopBases) | |
9 | tgcl 21577 | . . 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 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ↾t crest 16694 topGenctg 16711 Topctop 21501 TopBasesctb 21553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-bases 21554 |
This theorem is referenced by: resttopon 21769 resttopon2 21776 rest0 21777 restcld 21780 neitr 21788 restcls 21789 restntr 21790 ordtrest 21810 cmpsub 22008 fiuncmp 22012 1stcrest 22061 subislly 22089 llyrest 22093 nllyrest 22094 toplly 22098 cldllycmp 22103 kgencmp2 22154 llycmpkgen2 22158 1stckgen 22162 txkgen 22260 cnextfres1 22676 zdis 23424 cnmpopc 23532 dvbss 24499 dvreslem 24507 dvres2lem 24508 dvcnp2 24517 dvmptres 24560 ulmdvlem3 24990 psercn 25014 abelth 25029 ordtrestNEW 31164 cvxpconn 32489 cvmscld 32520 ptrest 34906 poimirlem29 34936 cnambfre 34955 limcresiooub 41972 limcresioolb 41973 cncfuni 42218 cncfiooicclem1 42225 fourierdlem32 42473 fourierdlem33 42474 fourierdlem48 42488 fourierdlem49 42489 fouriersw 42565 |
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