Theorem exrecfn

Description: Theorem about the existence of infinite recursive sets. y should usually be free in B . (Contributed by ML, 30-Mar-2022)

	Ref	Expression
Assertion	exrecfn	$$⊢\left(A\in V\wedge \forall yB\in W\right)\to \exists x\left(A\subseteq x\wedge \forall y\in xB\in x\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\left(z\in \mathrm{V}⟼z\cup \mathrm{ran}\left(y\in z⟼B\right)\right)=\left(z\in \mathrm{V}⟼z\cup \mathrm{ran}\left(y\in z⟼B\right)\right)$$
2	1	exrecfnlem	$$⊢\left(A\in V\wedge \forall yB\in W\right)\to \exists x\left(A\subseteq x\wedge \forall y\in xB\in x\right)$$