Theorem ovresd

Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	ovresd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	ovresd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
Assertion	ovresd	⊢ ( 𝜑 → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ovresd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
2		ovresd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
3		ovres	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )