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Mirrors > Home > MPE Home > Th. List > restuni | Structured version Visualization version GIF version |
Description: The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restuni.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
restuni | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | toptopon 21527 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | resttopon 21771 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
4 | 2, 3 | sylanb 583 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
5 | toponuni 21524 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) | |
6 | 4, 5 | syl 17 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 ‘cfv 6357 (class class class)co 7158 ↾t crest 16696 Topctop 21503 TopOnctopon 21520 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-en 8512 df-fin 8515 df-fi 8877 df-rest 16698 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 |
This theorem is referenced by: restuni2 21777 restcld 21782 restopn2 21787 neitr 21790 restcls 21791 restntr 21792 rncmp 22006 cmpsublem 22009 cmpsub 22010 fiuncmp 22014 connsubclo 22034 connima 22035 conncn 22036 nllyrest 22096 cldllycmp 22105 lly1stc 22106 llycmpkgen2 22160 1stckgen 22164 txkgen 22262 xkopjcn 22266 xkococnlem 22269 cnextfres1 22678 cnextfres 22679 cncfcnvcn 23531 cnheibor 23561 evthicc 24062 psercn 25016 abelth 25031 connpconn 32484 cvmscld 32522 cvmsss2 32523 cvmliftmolem1 32530 cvmliftlem10 32543 cvmlift2lem9 32560 cvmlift2lem11 32562 cvmlift2lem12 32563 cvmlift3lem7 32574 ivthALT 33685 ptrest 34893 poimirlem29 34923 poimirlem30 34924 poimirlem31 34925 poimir 34927 cncfuni 42176 cncfiooicclem1 42183 stoweidlem28 42320 dirkercncflem4 42398 fourierdlem42 42441 |
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