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Metamath Proof Explorer

Theorem fnssresd

Description: Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Hypotheses fnssresd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
fnssresd.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion fnssresd ( 𝜑 → ( 𝐹 ↾ 𝐵 ) Fn 𝐵 )

Proof

Step Hyp Ref Expression
1 fnssresd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2 fnssresd.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3 fnssres ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐵 ) Fn 𝐵 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐹 ↾ 𝐵 ) Fn 𝐵 )