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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 22910 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | resttopon 23156 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
4 | 2, 3 | sylanb 579 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
5 | toponuni 22907 | . 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 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ∪ cuni 4913 ‘cfv 6554 (class class class)co 7424 ↾t crest 17435 Topctop 22886 TopOnctopon 22903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-en 8975 df-fin 8978 df-fi 9454 df-rest 17437 df-topgen 17458 df-top 22887 df-topon 22904 df-bases 22940 |
This theorem is referenced by: restuni2 23162 restcld 23167 restopn2 23172 neitr 23175 restcls 23176 restntr 23177 rncmp 23391 cmpsublem 23394 cmpsub 23395 fiuncmp 23399 connsubclo 23419 connima 23420 conncn 23421 nllyrest 23481 cldllycmp 23490 lly1stc 23491 llycmpkgen2 23545 1stckgen 23549 txkgen 23647 xkopjcn 23651 xkococnlem 23654 cnextfres1 24063 cnextfres 24064 cncfcnvcn 24937 cnheibor 24972 evthicc 25479 psercn 26456 abelth 26471 zarmxt1 33695 connpconn 35063 cvmscld 35101 cvmsss2 35102 cvmliftmolem1 35109 cvmliftlem10 35122 cvmlift2lem9 35139 cvmlift2lem11 35141 cvmlift2lem12 35142 cvmlift3lem7 35153 ivthALT 36047 ptrest 37320 poimirlem29 37350 poimirlem30 37351 poimirlem31 37352 poimir 37354 cncfuni 45507 cncfiooicclem1 45514 stoweidlem28 45649 dirkercncflem4 45727 fourierdlem42 45770 restcls2lem 48246 iscnrm3rlem7 48280 |
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