Theorem strss

Description: Propagate component extraction to a structure T from a subset structure S . (Contributed by Mario Carneiro, 11-Oct-2013) (Revised by Mario Carneiro, 15-Jan-2014)

	Ref	Expression
Hypotheses	strss.t	⊢ 𝑇 ∈ V
	strss.f	⊢ Fun 𝑇
	strss.s	⊢ 𝑆 ⊆ 𝑇
	strss.e	⊢ 𝐸 = Slot ( 𝐸 ‘ ndx )
	strss.n	⊢ ⟨ ( 𝐸 ‘ ndx ) , 𝐶 ⟩ ∈ 𝑆
Assertion	strss	⊢ ( 𝐸 ‘ 𝑇 ) = ( 𝐸 ‘ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		strss.t	⊢ 𝑇 ∈ V
2		strss.f	⊢ Fun 𝑇
3		strss.s	⊢ 𝑆 ⊆ 𝑇
4		strss.e	⊢ 𝐸 = Slot ( 𝐸 ‘ ndx )
5		strss.n	⊢ ⟨ ( 𝐸 ‘ ndx ) , 𝐶 ⟩ ∈ 𝑆
6	1	a1i	⊢ ( ⊤ → 𝑇 ∈ V )
7	2	a1i	⊢ ( ⊤ → Fun 𝑇 )
8	3	a1i	⊢ ( ⊤ → 𝑆 ⊆ 𝑇 )
9	5	a1i	⊢ ( ⊤ → ⟨ ( 𝐸 ‘ ndx ) , 𝐶 ⟩ ∈ 𝑆 )
10	4 6 7 8 9	strssd	⊢ ( ⊤ → ( 𝐸 ‘ 𝑇 ) = ( 𝐸 ‘ 𝑆 ) )
11	10	mptru	⊢ ( 𝐸 ‘ 𝑇 ) = ( 𝐸 ‘ 𝑆 )