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Theorem isxms 22374
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 6304 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
2 isms.j . . . 4 𝐽 = (TopOpen‘𝐾)
31, 2syl6eqr 2776 . . 3 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
4 fveq2 6304 . . . . . 6 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
5 fveq2 6304 . . . . . . . 8 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
6 isms.x . . . . . . . 8 𝑋 = (Base‘𝐾)
75, 6syl6eqr 2776 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
87sqxpeqd 5250 . . . . . 6 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
94, 8reseq12d 5504 . . . . 5 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
10 isms.d . . . . 5 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
119, 10syl6eqr 2776 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
1211fveq2d 6308 . . 3 (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷))
133, 12eqeq12d 2739 . 2 (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷)))
14 df-xms 22247 . 2 ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
1513, 14elrab2 3472 1 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1596  wcel 2103   × cxp 5216  cres 5220  cfv 6001  Basecbs 15980  distcds 16073  TopOpenctopn 16205  MetOpencmopn 19859  TopSpctps 20859  ∞MetSpcxme 22244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-res 5230  df-iota 5964  df-fv 6009  df-xms 22247
This theorem is referenced by:  isxms2  22375  xmstopn  22378  xmstps  22380  xmspropd  22400
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