Theorem wfr2

Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of F at any X e. A is G recursively applied to all "previous" values of F . (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr2.1	$$⊢R\mathrm{We}A$$
	wfr2.2	$$⊢R\mathrm{Se}A$$
	wfr2.3	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
Assertion	wfr2	$$⊢X\in A\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$

Proof

Step	Hyp	Ref	Expression
1		wfr2.1	$$⊢R\mathrm{We}A$$
2		wfr2.2	$$⊢R\mathrm{Se}A$$
3		wfr2.3	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
4		eqid	$$⊢F\cup \left\{⟨x,G\left({F\upharpoonright }_{Pred\left(R,A,x\right)}\right)⟩\right\}=F\cup \left\{⟨x,G\left({F\upharpoonright }_{Pred\left(R,A,x\right)}\right)⟩\right\}$$
5	1 2 3 4	wfrlem16	$$⊢\mathrm{dom}F=A$$
6	5	eleq2i	$$⊢X\in \mathrm{dom}F\leftrightarrow X\in A$$
7	1 2 3	wfr2a	$$⊢X\in \mathrm{dom}F\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$
8	6 7	sylbir	$$⊢X\in A\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$