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Theorem isxms 12630
Description: Express the predicate "𝑋, 𝐷 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
isxms (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))

Proof of Theorem isxms
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq2 5421 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
2 isms.j . . . 4 𝐽 = (TopOpen‘𝐾)
31, 2syl6eqr 2190 . . 3 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
4 fveq2 5421 . . . . . 6 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
5 fveq2 5421 . . . . . . . 8 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
6 isms.x . . . . . . . 8 𝑋 = (Base‘𝐾)
75, 6syl6eqr 2190 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
87sqxpeqd 4565 . . . . . 6 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
94, 8reseq12d 4820 . . . . 5 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
10 isms.d . . . . 5 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
119, 10syl6eqr 2190 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
1211fveq2d 5425 . . 3 (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷))
133, 12eqeq12d 2154 . 2 (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷)))
14 df-xms 12518 . 2 ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
1513, 14elrab2 2843 1 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1331  wcel 1480   × cxp 4537  cres 4541  cfv 5123  Basecbs 11969  distcds 12040  TopOpenctopn 12131  MetOpencmopn 12164  TopSpctps 12207  ∞MetSpcxms 12515
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-xms 12518
This theorem is referenced by:  isxms2  12631  xmstopn  12634  xmstps  12636  xmspropd  12656
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