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Mirrors > Home > ILE Home > Th. List > isms | GIF version |
Description: Express the predicate "〈𝑋, 𝐷〉 is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
isms | ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5421 | . . . . 5 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
2 | fveq2 5421 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
3 | isms.x | . . . . . . 7 ⊢ 𝑋 = (Base‘𝐾) | |
4 | 2, 3 | syl6eqr 2190 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
5 | 4 | sqxpeqd 4565 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
6 | 1, 5 | reseq12d 4820 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋))) |
7 | isms.d | . . . 4 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
8 | 6, 7 | syl6eqr 2190 | . . 3 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
9 | 4 | fveq2d 5425 | . . 3 ⊢ (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋)) |
10 | 8, 9 | eleq12d 2210 | . 2 ⊢ (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋))) |
11 | df-ms 12509 | . 2 ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))} | |
12 | 10, 11 | elrab2 2843 | 1 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 × cxp 4537 ↾ cres 4541 ‘cfv 5123 Basecbs 11959 distcds 12030 TopOpenctopn 12121 Metcmet 12150 ∞MetSpcxms 12505 MetSpcms 12506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-res 4551 df-iota 5088 df-fv 5131 df-ms 12509 |
This theorem is referenced by: isms2 12623 msxms 12627 mspropd 12647 |
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