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Metamath Proof Explorer

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 ) )