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Mirrors > Home > MPE Home > Th. List > setsmstset | Structured version Visualization version GIF version |
Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
setsms.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
Ref | Expression |
---|---|
setsmstset | ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
2 | fvex 6935 | . . 3 ⊢ (MetOpen‘𝐷) ∈ V | |
3 | tsetid 17414 | . . . 4 ⊢ TopSet = Slot (TopSet‘ndx) | |
4 | 3 | setsid 17257 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (MetOpen‘𝐷) ∈ V) → (MetOpen‘𝐷) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
5 | 1, 2, 4 | sylancl 585 | . 2 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
6 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
7 | 6 | fveq2d 6926 | . 2 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
8 | 5, 7 | eqtr4d 2783 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 〈cop 4654 × cxp 5698 ↾ cres 5702 ‘cfv 6575 (class class class)co 7450 sSet csts 17212 ndxcnx 17242 Basecbs 17260 TopSetcts 17319 distcds 17322 MetOpencmopn 21379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-1cn 11244 ax-addcl 11246 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-sets 17213 df-slot 17231 df-ndx 17243 df-tset 17332 |
This theorem is referenced by: setsmstopn 24513 |
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