Theorem wfr2OLD

Description: Obsolete version of wfr2 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr2OLD.1	$$⊢R\mathrm{We}A$$
	wfr2OLD.2	$$⊢R\mathrm{Se}A$$
	wfr2OLD.3	$$⊢F=\mathrm{wrecs}\left(R,A,G\right)$$
Assertion	wfr2OLD	$$⊢X\in A\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$

Proof

Step	Hyp	Ref	Expression
1		wfr2OLD.1	$$⊢R\mathrm{We}A$$
2		wfr2OLD.2	$$⊢R\mathrm{Se}A$$
3		wfr2OLD.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	wfrlem16OLD	$$⊢\mathrm{dom}F=A$$
6	5	eleq2i	$$⊢X\in \mathrm{dom}F\leftrightarrow X\in A$$
7	1 2 3	wfr2aOLD	$$⊢X\in \mathrm{dom}F\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$
8	6 7	sylbir	$$⊢X\in A\to F\left(X\right)=G\left({F\upharpoonright }_{Pred\left(R,A,X\right)}\right)$$