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 = wrecs ( R , A , G )
Assertion	wfrdmcl	|- ( X e. dom F -> Pred ( R , A , X ) C_ dom F )

Proof

Step	Hyp	Ref	Expression
1		wfrrel.1	|- F = wrecs ( R , A , G )
2		df-wrecs	|- wrecs ( R , A , G ) = frecs ( R , A , ( G o. 2nd ) )
3	1 2	eqtri	|- F = frecs ( R , A , ( G o. 2nd ) )
4	3	frrdmcl	|- ( X e. dom F -> Pred ( R , A , X ) C_ dom F )