Theorem psmetdmdm

Description: Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018)

	Ref	Expression
Assertion	psmetdmdm	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝑋 = dom dom 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		elfvex	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝑋 ∈ V )
2		ispsmet	⊢ ( 𝑋 ∈ V → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐷 𝑥 ) = 0 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) ) )
3	2	biimpa	⊢ ( ( 𝑋 ∈ V ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐷 𝑥 ) = 0 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) )
4	1 3	mpancom	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐷 𝑥 ) = 0 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) )
5	4	simpld	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* )
6		fdm	⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → dom 𝐷 = ( 𝑋 × 𝑋 ) )
7	6	dmeqd	⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → dom dom 𝐷 = dom ( 𝑋 × 𝑋 ) )
8	5 7	syl	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → dom dom 𝐷 = dom ( 𝑋 × 𝑋 ) )
9		dmxpid	⊢ dom ( 𝑋 × 𝑋 ) = 𝑋
10	8 9	eqtr2di	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝑋 = dom dom 𝐷 )