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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 6932 | . . 3 ⊢ (MetOpen‘𝐷) ∈ V | |
3 | tsetid 17407 | . . . 4 ⊢ TopSet = Slot (TopSet‘ndx) | |
4 | 3 | setsid 17250 | . . 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 6923 | . 2 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
8 | 5, 7 | eqtr4d 2777 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 Vcvv 3482 〈cop 4654 × cxp 5697 ↾ cres 5701 ‘cfv 6572 (class class class)co 7445 sSet csts 17205 ndxcnx 17235 Basecbs 17253 TopSetcts 17312 distcds 17315 MetOpencmopn 21372 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-1cn 11238 ax-addcl 11240 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-sets 17206 df-slot 17224 df-ndx 17236 df-tset 17325 |
This theorem is referenced by: setsmstopn 24504 |
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