Theorem coe1fval

Description: Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015)

	Ref	Expression
Hypothesis	coe1fval.a	⊢ 𝐴 = ( coe1 ‘ 𝐹 )
Assertion	coe1fval	⊢ ( 𝐹 ∈ 𝑉 → 𝐴 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		coe1fval.a	⊢ 𝐴 = ( coe1 ‘ 𝐹 )
2		elex	⊢ ( 𝐹 ∈ 𝑉 → 𝐹 ∈ V )
3		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 1o × { 𝑛 } ) ) = ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) )
4	3	mpteq2dv	⊢ ( 𝑓 = 𝐹 → ( 𝑛 ∈ ℕ0 ↦ ( 𝑓 ‘ ( 1o × { 𝑛 } ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) )
5		df-coe1	⊢ coe1 = ( 𝑓 ∈ V ↦ ( 𝑛 ∈ ℕ0 ↦ ( 𝑓 ‘ ( 1o × { 𝑛 } ) ) ) )
6		nn0ex	⊢ ℕ0 ∈ V
7	6	mptex	⊢ ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) ∈ V
8	4 5 7	fvmpt	⊢ ( 𝐹 ∈ V → ( coe1 ‘ 𝐹 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) )
9	1 8	eqtrid	⊢ ( 𝐹 ∈ V → 𝐴 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) )
10	2 9	syl	⊢ ( 𝐹 ∈ 𝑉 → 𝐴 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) )