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Metamath Proof Explorer

Theorem resabs1d

Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypothesis resabs1d.b ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
Assertion resabs1d ( 𝜑 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 resabs1d.b ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
2 resabs1 ⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) )
3 1 2 syl ⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) )