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Metamath Proof Explorer

Theorem resabs2d

Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypothesis resabs2d.1 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
Assertion resabs2d ( 𝜑 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 resabs2d.1 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
2 resabs2 ⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )
3 1 2 syl ⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )