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Metamath Proof Explorer

Theorem fssd

Description: Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fssd.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
fssd.b ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
Assertion fssd ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )

Proof

Step Hyp Ref Expression
1 fssd.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2 fssd.b ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3 fss ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 ⟶ 𝐶 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )