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Mirrors > Home > ILE Home > Th. List > setsmstsetg | GIF version |
Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.) |
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)〉)) |
setsmsbasg.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
setsmsbasg.d | ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊) |
Ref | Expression |
---|---|
setsmstsetg | ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsmsbasg.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
2 | setsmsbasg.d | . . 3 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊) | |
3 | tsetslid 12779 | . . . 4 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
4 | 3 | setsslid 12643 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (MetOpen‘𝐷) ∈ 𝑊) → (MetOpen‘𝐷) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
5 | 1, 2, 4 | syl2anc 411 | . 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 5546 | . 2 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
8 | 5, 7 | eqtr4d 2225 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 〈cop 3617 × cxp 4649 ↾ cres 4653 ‘cfv 5242 (class class class)co 5906 ndxcnx 12589 sSet csts 12590 Basecbs 12592 TopSetcts 12675 distcds 12678 MetOpencmopn 14001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1re 7952 ax-addrcl 7955 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-iota 5203 df-fun 5244 df-fv 5250 df-ov 5909 df-oprab 5910 df-mpo 5911 df-inn 8969 df-2 9027 df-3 9028 df-4 9029 df-5 9030 df-6 9031 df-7 9032 df-8 9033 df-9 9034 df-ndx 12595 df-slot 12596 df-sets 12599 df-tset 12688 |
This theorem is referenced by: (None) |
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