Theorem isms 24344

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 6861	. . . . 5 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
2		fveq2 6861	. . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
3		isms.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐾)
4	2, 3	eqtr4di 2783	. . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
5	4	sqxpeqd 5673	. . . . 5 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
6	1, 5	reseq12d 5954	. . . 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 2783	. . 3 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
9	4	fveq2d 6865	. . 3 ⊢ (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
10	8, 9	eleq12d 2823	. 2 ⊢ (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋)))
11		df-ms 24216	. 2 ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
12	10, 11	elrab2 3665	1 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   × cxp 5639   ↾ cres 5643  ‘cfv 6514  Basecbs 17186  distcds 17236  TopOpenctopn 17391  Metcmet 21257  ∞MetSpcxms 24212  MetSpcms 24213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-res 5653  df-iota 6467  df-fv 6522  df-ms 24216

This theorem is referenced by:  isms2  24345  msxms  24349  mspropd  24369  setsms  24375  tmsms  24382  imasf1oms  24385  ressms  24421  prdsms  24426