Theorem strfvn

Description: Value of a structure component extractor E . Normally, E is a defined constant symbol such as Base ( df-base ) and N is a fixed integer such as 1 . S is a structure, i.e. a specific member of a class of structures such as Poset ( df-poset ) where S e. Poset .

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv . (Contributed by NM, 9-Sep-2011) (Revised by Mario Carneiro, 6-Oct-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	strfvn.f	⊢ 𝑆 ∈ V
	strfvn.c	⊢ 𝐸 = Slot 𝑁
Assertion	strfvn	⊢ ( 𝐸 ‘ 𝑆 ) = ( 𝑆 ‘ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		strfvn.f	⊢ 𝑆 ∈ V
2		strfvn.c	⊢ 𝐸 = Slot 𝑁
3	1	a1i	⊢ ( ⊤ → 𝑆 ∈ V )
4	2 3	strfvnd	⊢ ( ⊤ → ( 𝐸 ‘ 𝑆 ) = ( 𝑆 ‘ 𝑁 ) )
5	4	mptru	⊢ ( 𝐸 ‘ 𝑆 ) = ( 𝑆 ‘ 𝑁 )