Theorem isms 24774

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.

Step	Hyp	Ref	Expression
1		fveq2 6886	. . . . 5 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
2		fveq2 6886	. . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
3		isms.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐾)
4	2, 3	eqtr4di 2813	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
5	4	sqxpeqd 5682	. . . . 5 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
6	1, 5	reseq12d 5970	. . . 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	eqtr4di 2813	. . 3 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
9	4	fveq2d 6890	. . 3 ⊢ (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
10	8, 9	eleq12d 2854	. 2 ⊢ (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋)))
11		df-ms 24646	. 2 ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
12	10, 11	elrab2 3648	1 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   × cxp 5648   ↾ cres 5652  ‘cfv 6539  Basecbs 17393  distcds 17443  TopOpenctopn 17598  Metcmet 21670  ∞MetSpcxms 24642  MetSpcms 24643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-res 5662  df-iota 6494  df-fv 6547  df-ms 24646

This theorem is used by:  isms2  24775  msxms  24779  mspropd  24799  setsms  24805  tmsms  24812  imasf1oms  24815  ressms  24851  prdsms  24856