Theorem wfrdmcl

Description: The predecessor class of an element of the well-ordered recursion generator's domain is a subset of its domain. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011) (Proof shortened by Scott Fenton, 17-Nov-2024)

	Ref	Expression
Hypothesis	wfrrel.1	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
Assertion	wfrdmcl	$$⊢X\in \mathrm{dom}F\to Pred\left(R,A,X\right)\subseteq \mathrm{dom}F$$

Proof

Step	Hyp	Ref	Expression
1		wfrrel.1	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
2		df-wrecs	$$⊢\mathrm{wrecs}\left(R,A,G\right)=\mathrm{frecs}\left(R,A,G\circ 2^{nd}\right)$$
3	1 2	eqtri	$$⊢F=\mathrm{frecs}\left(R,A,G\circ 2^{nd}\right)$$
4	3	frrdmcl	$$⊢X\in \mathrm{dom}F\to Pred\left(R,A,X\right)\subseteq \mathrm{dom}F$$