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Metamath Proof Explorer

Theorem psmetf

Description: The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018)

Ref Expression
Assertion psmetf ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* )

Proof

Step Hyp Ref Expression
1 elfvex ⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝑋 ∈ V )
2 ispsmet ⊢ ( 𝑋 ∈ V → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +𝑒 ( 𝑐 𝐷 𝑏 ) ) ) ) ) )
3 1 2 syl ⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +𝑒 ( 𝑐 𝐷 𝑏 ) ) ) ) ) )
4 3 ibi ⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +𝑒 ( 𝑐 𝐷 𝑏 ) ) ) ) )
5 4 simpld ⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* )