Theorem isxms 24430

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.

Step	Hyp	Ref	Expression
1		fveq2 6827	. . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
2		isms.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐾)
3	1, 2	eqtr4di 2792	. . 3 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
4		fveq2 6827	. . . . . 6 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
5		fveq2 6827	. . . . . . . 8 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
6		isms.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐾)
7	5, 6	eqtr4di 2792	. . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
8	7	sqxpeqd 5650	. . . . . 6 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
9	4, 8	reseq12d 5932	. . . . 5 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
10		isms.d	. . . . 5 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
11	9, 10	eqtr4di 2792	. . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
12	11	fveq2d 6831	. . 3 ⊢ (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷))
13	3, 12	eqeq12d 2755	. 2 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷)))
14		df-xms 24303	. 2 ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
15	13, 14	elrab2 3632	1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   × cxp 5616   ↾ cres 5620  ‘cfv 6485  Basecbs 17170  distcds 17220  TopOpenctopn 17375  MetOpencmopn 21337  TopSpctps 22915  ∞MetSpcxms 24300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-res 5630  df-iota 6441  df-fv 6493  df-xms 24303

This theorem is referenced by:  isxms2  24431  xmstopn  24434  xmstps  24436  xmspropd  24456