Theorem tfrlem6

Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994) (Revised by Mario Carneiro, 9-May-2015) Avoid ax-10 , ax-nul , ax-pr , ax-sep and ax-un . (Revised by Umit Teoman Dogan, 10-Jun-2026)

	Ref	Expression
Hypothesis	tfrlem.1	⊢ 𝐴 = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) }
Assertion	tfrlem6	⊢ Rel recs ( 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		tfrlem.1	⊢ 𝐴 = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) }
2		df-recs	⊢ recs ( 𝐹 ) = wrecs ( E , On , 𝐹 )
3	2	wfrrel	⊢ Rel recs ( 𝐹 )