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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 6676 | . . 3 ⊢ (MetOpen‘𝐷) ∈ V | |
3 | tsetid 16648 | . . . 4 ⊢ TopSet = Slot (TopSet‘ndx) | |
4 | 3 | setsid 16526 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (MetOpen‘𝐷) ∈ V) → (MetOpen‘𝐷) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
5 | 1, 2, 4 | sylancl 586 | . 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 6667 | . 2 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
8 | 5, 7 | eqtr4d 2856 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 × cxp 5546 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ndxcnx 16468 sSet csts 16469 Basecbs 16471 TopSetcts 16559 distcds 16562 MetOpencmopn 20463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addcl 10585 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-ndx 16474 df-slot 16475 df-sets 16478 df-tset 16572 |
This theorem is referenced by: setsmstopn 23015 |
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