Theorem wfr1

Description: The Principle of Well-Ordered 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-ordered 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	$$⊢R\mathrm{We}A$$
	wfr1.2	$$⊢R\mathrm{Se}A$$
	wfr1.3	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
Assertion	wfr1	$$⊢FFnA$$

Proof

Step	Hyp	Ref	Expression
1		wfr1.1	$$⊢R\mathrm{We}A$$
2		wfr1.2	$$⊢R\mathrm{Se}A$$
3		wfr1.3	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
4	1 2 3	wfrfun	$$⊢\mathrm{Fun}F$$
5		eqid	$$⊢F\cup \left\{⟨z,G\left({F\upharpoonright }_{Pred\left(R,A,z\right)}\right)⟩\right\}=F\cup \left\{⟨z,G\left({F\upharpoonright }_{Pred\left(R,A,z\right)}\right)⟩\right\}$$
6	1 2 3 5	wfrlem16	$$⊢\mathrm{dom}F=A$$
7		df-fn	$$⊢FFnA\leftrightarrow \left(\mathrm{Fun}F\wedge \mathrm{dom}F=A\right)$$
8	4 6 7	mpbir2an	$$⊢FFnA$$