Theorem funfni

Description: Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004)

	Ref	Expression
Hypothesis	funfni.1	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → 𝜑 )
Assertion	funfni	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		funfni.1	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → 𝜑 )
2		fnfun	⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 )
3		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
4	3	eleq2d	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴 ) )
5	4	biimpar	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ dom 𝐹 )
6	2 5 1	syl2an2r	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝜑 )