Theorem rdg0

Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995) (Revised by Mario Carneiro, 14-Nov-2014)

	Ref	Expression
Hypothesis	rdg.1	$$⊢A\in \mathrm{V}$$
Assertion	rdg0	$$⊢\mathrm{rec}\left(F,A\right)\left(\varnothing \right)=A$$

Proof

Step	Hyp	Ref	Expression
1		rdg.1	$$⊢A\in \mathrm{V}$$
2		rdgdmlim	$$⊢\mathrm{Lim}\mathrm{dom}\mathrm{rec}\left(F,A\right)$$
3		limomss	$$⊢\mathrm{Lim}\mathrm{dom}\mathrm{rec}\left(F,A\right)\to \mathrm{\omega }\subseteq \mathrm{dom}\mathrm{rec}\left(F,A\right)$$
4	2 3	ax-mp	$$⊢\mathrm{\omega }\subseteq \mathrm{dom}\mathrm{rec}\left(F,A\right)$$
5		peano1	$$⊢\varnothing \in \mathrm{\omega }$$
6	4 5	sselii	$$⊢\varnothing \in \mathrm{dom}\mathrm{rec}\left(F,A\right)$$
7		eqid	$$⊢\left(x\in \mathrm{V}⟼if\left(x=\varnothing ,A,if\left(\mathrm{Lim}\mathrm{dom}x,\bigcup \mathrm{ran}x,F\left(x\left(\bigcup \mathrm{dom}x\right)\right)\right)\right)\right)=\left(x\in \mathrm{V}⟼if\left(x=\varnothing ,A,if\left(\mathrm{Lim}\mathrm{dom}x,\bigcup \mathrm{ran}x,F\left(x\left(\bigcup \mathrm{dom}x\right)\right)\right)\right)\right)$$
8		rdgvalg	$$⊢y\in \mathrm{dom}\mathrm{rec}\left(F,A\right)\to \mathrm{rec}\left(F,A\right)\left(y\right)=\left(x\in \mathrm{V}⟼if\left(x=\varnothing ,A,if\left(\mathrm{Lim}\mathrm{dom}x,\bigcup \mathrm{ran}x,F\left(x\left(\bigcup \mathrm{dom}x\right)\right)\right)\right)\right)\left({\mathrm{rec}\left(F,A\right)\upharpoonright }_y\right)$$
9	7 8 1	tz7.44-1	$$⊢\varnothing \in \mathrm{dom}\mathrm{rec}\left(F,A\right)\to \mathrm{rec}\left(F,A\right)\left(\varnothing \right)=A$$
10	6 9	ax-mp	$$⊢\mathrm{rec}\left(F,A\right)\left(\varnothing \right)=A$$