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Theorem isms 12622
 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.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
isms (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))

Proof of Theorem isms
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef 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)
42, 3syl6eqr 2190 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
54sqxpeqd 4565 . . . . 5 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
61, 5reseq12d 4820 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
7 isms.d . . . 4 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
86, 7syl6eqr 2190 . . 3 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
94fveq2d 5425 . . 3 (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
108, 9eleq12d 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))}
1210, 11elrab2 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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