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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 22923 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
| 3 | resttopon 23169 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
| 4 | 2, 3 | sylanb 581 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
| 5 | toponuni 22920 | . 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 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 Topctop 22899 TopOnctopon 22916 |
| 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-topon 22917 df-bases 22953 |
| This theorem is referenced by: restuni2 23175 restcld 23180 restopn2 23185 neitr 23188 restcls 23189 restntr 23190 rncmp 23404 cmpsublem 23407 cmpsub 23408 fiuncmp 23412 connsubclo 23432 connima 23433 conncn 23434 nllyrest 23494 cldllycmp 23503 lly1stc 23504 llycmpkgen2 23558 1stckgen 23562 txkgen 23660 xkopjcn 23664 xkococnlem 23667 cnextfres1 24076 cnextfres 24077 cncfcnvcn 24952 cnheibor 24987 evthicc 25494 psercn 26470 abelth 26485 zarmxt1 33879 connpconn 35240 cvmscld 35278 cvmsss2 35279 cvmliftmolem1 35286 cvmliftlem10 35299 cvmlift2lem9 35316 cvmlift2lem11 35318 cvmlift2lem12 35319 cvmlift3lem7 35330 ivthALT 36336 ptrest 37626 poimirlem29 37656 poimirlem30 37657 poimirlem31 37658 poimir 37660 cncfuni 45901 cncfiooicclem1 45908 stoweidlem28 46043 dirkercncflem4 46121 fourierdlem42 46164 restcls2lem 48810 iscnrm3rlem7 48843 |
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