Theorem wfr1

Description: The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function G . Then, using a base class A and a well-ordering R of A , we define a function F . This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of F . We begin by showing that F is a function over A . (Contributed by Scott Fenton, 22-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr1.1	|- R We A
	wfr1.2	|- R Se A
	wfr1.3	|- F = wrecs ( R , A , G )
Assertion	wfr1	|- F Fn A

Proof

Step	Hyp	Ref	Expression
1		wfr1.1	|- R We A
2		wfr1.2	|- R Se A
3		wfr1.3	|- F = wrecs ( R , A , G )
4	1 2 3	wfrfun	|- Fun F
5		eqid	|- ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
6	1 2 3 5	wfrlem16	|- dom F = A
7		df-fn	|- ( F Fn A <-> ( Fun F /\ dom F = A ) )
8	4 6 7	mpbir2an	|- F Fn A