Theorem wfr2

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

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

Proof

Step	Hyp	Ref	Expression
1		wfr2.1	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
2	1	wfr1	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to FFnA$$
3	2	fndmd	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to \mathrm{dom}F=A$$
4	3	eleq2d	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to \left(X\in \mathrm{dom}F\leftrightarrow X\in A\right)$$
5	4	biimpar	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge X\in A\right)\to X\in \mathrm{dom}F$$
6	1	wfr2a	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge X\in \mathrm{dom}F\right)\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$
7	5 6	syldan	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge X\in A\right)\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$