Theorem restabs

Description: Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009) (Proof shortened by Mario Carneiro, 1-May-2015)

	Ref	Expression
Assertion	restabs	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → ( ( 𝐽 ↾t 𝑇 ) ↾t 𝑆 ) = ( 𝐽 ↾t 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → 𝐽 ∈ 𝑉 )
2		simp3	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → 𝑇 ∈ 𝑊 )
3		ssexg	⊢ ( ( 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → 𝑆 ∈ V )
4	3	3adant1	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → 𝑆 ∈ V )
5		restco	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊 ∧ 𝑆 ∈ V ) → ( ( 𝐽 ↾t 𝑇 ) ↾t 𝑆 ) = ( 𝐽 ↾t ( 𝑇 ∩ 𝑆 ) ) )
6	1 2 4 5	syl3anc	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → ( ( 𝐽 ↾t 𝑇 ) ↾t 𝑆 ) = ( 𝐽 ↾t ( 𝑇 ∩ 𝑆 ) ) )
7		simp2	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → 𝑆 ⊆ 𝑇 )
8		sseqin2	⊢ ( 𝑆 ⊆ 𝑇 ↔ ( 𝑇 ∩ 𝑆 ) = 𝑆 )
9	7 8	sylib	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → ( 𝑇 ∩ 𝑆 ) = 𝑆 )
10	9	oveq2d	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → ( 𝐽 ↾t ( 𝑇 ∩ 𝑆 ) ) = ( 𝐽 ↾t 𝑆 ) )
11	6 10	eqtrd	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊 ) → ( ( 𝐽 ↾t 𝑇 ) ↾t 𝑆 ) = ( 𝐽 ↾t 𝑆 ) )