Theorem nfwrecs

Description: Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018) (Proof shortened by Scott Fenton, 17-Nov-2024)

	Ref	Expression
Hypotheses	nfwrecs.1	⊢ Ⅎ 𝑥 𝑅
	nfwrecs.2	⊢ Ⅎ 𝑥 𝐴
	nfwrecs.3	⊢ Ⅎ 𝑥 𝐹
Assertion	nfwrecs	⊢ Ⅎ 𝑥 wrecs ( 𝑅 , 𝐴 , 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		nfwrecs.1	⊢ Ⅎ 𝑥 𝑅
2		nfwrecs.2	⊢ Ⅎ 𝑥 𝐴
3		nfwrecs.3	⊢ Ⅎ 𝑥 𝐹
4		df-wrecs	⊢ wrecs ( 𝑅 , 𝐴 , 𝐹 ) = frecs ( 𝑅 , 𝐴 , ( 𝐹 ∘ 2nd ) )
5		nfcv	⊢ Ⅎ 𝑥 2nd
6	3 5	nfco	⊢ Ⅎ 𝑥 ( 𝐹 ∘ 2nd )
7	1 2 6	nffrecs	⊢ Ⅎ 𝑥 frecs ( 𝑅 , 𝐴 , ( 𝐹 ∘ 2nd ) )
8	4 7	nfcxfr	⊢ Ⅎ 𝑥 wrecs ( 𝑅 , 𝐴 , 𝐹 )