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 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 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 ∘ 2nd ) )
3	1 2	eqtri	⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , ( 𝐺 ∘ 2nd ) )
4	3	frrdmcl	⊢ ( 𝑋 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝐹 )