Theorem fr0g

Description: The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996)

	Ref	Expression
Assertion	fr0g	⊢ ( 𝐴 ∈ 𝐵 → ( ( rec ( 𝐹 , 𝐴 ) ↾ ω ) ‘ ∅ ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		peano1	⊢ ∅ ∈ ω
2		fvres	⊢ ( ∅ ∈ ω → ( ( rec ( 𝐹 , 𝐴 ) ↾ ω ) ‘ ∅ ) = ( rec ( 𝐹 , 𝐴 ) ‘ ∅ ) )
3	1 2	ax-mp	⊢ ( ( rec ( 𝐹 , 𝐴 ) ↾ ω ) ‘ ∅ ) = ( rec ( 𝐹 , 𝐴 ) ‘ ∅ )
4		rdg0g	⊢ ( 𝐴 ∈ 𝐵 → ( rec ( 𝐹 , 𝐴 ) ‘ ∅ ) = 𝐴 )
5	3 4	eqtrid	⊢ ( 𝐴 ∈ 𝐵 → ( ( rec ( 𝐹 , 𝐴 ) ↾ ω ) ‘ ∅ ) = 𝐴 )