Theorem rdglimg

Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014)

	Ref	Expression
Assertion	rdglimg	$$⊢\left(B\in \mathrm{dom}\mathrm{rec}\left(F,A\right)\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup \mathrm{rec}\left(F,A\right)\left[B\right]$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\left(x\in \mathrm{V}⟼if\left(x=\varnothing ,A,if\left(\mathrm{Lim}\mathrm{dom}x,\bigcup \mathrm{ran}x,F\left(x\left(\bigcup \mathrm{dom}x\right)\right)\right)\right)\right)=\left(x\in \mathrm{V}⟼if\left(x=\varnothing ,A,if\left(\mathrm{Lim}\mathrm{dom}x,\bigcup \mathrm{ran}x,F\left(x\left(\bigcup \mathrm{dom}x\right)\right)\right)\right)\right)$$
2		rdgvalg	$$⊢y\in \mathrm{dom}\mathrm{rec}\left(F,A\right)\to \mathrm{rec}\left(F,A\right)\left(y\right)=\left(x\in \mathrm{V}⟼if\left(x=\varnothing ,A,if\left(\mathrm{Lim}\mathrm{dom}x,\bigcup \mathrm{ran}x,F\left(x\left(\bigcup \mathrm{dom}x\right)\right)\right)\right)\right)\left({\mathrm{rec}\left(F,A\right)\upharpoonright }_y\right)$$
3		rdgseg	$$⊢y\in \mathrm{dom}\mathrm{rec}\left(F,A\right)\to {\mathrm{rec}\left(F,A\right)\upharpoonright }_y\in \mathrm{V}$$
4		rdgfun	$$⊢\mathrm{Fun}\mathrm{rec}\left(F,A\right)$$
5		funfn	$$⊢\mathrm{Fun}\mathrm{rec}\left(F,A\right)\leftrightarrow \mathrm{rec}\left(F,A\right)Fn\mathrm{dom}\mathrm{rec}\left(F,A\right)$$
6	4 5	mpbi	$$⊢\mathrm{rec}\left(F,A\right)Fn\mathrm{dom}\mathrm{rec}\left(F,A\right)$$
7		rdgdmlim	$$⊢\mathrm{Lim}\mathrm{dom}\mathrm{rec}\left(F,A\right)$$
8		limord	$$⊢\mathrm{Lim}\mathrm{dom}\mathrm{rec}\left(F,A\right)\to \mathrm{Ord}\mathrm{dom}\mathrm{rec}\left(F,A\right)$$
9	7 8	ax-mp	$$⊢\mathrm{Ord}\mathrm{dom}\mathrm{rec}\left(F,A\right)$$
10	1 2 3 6 9	tz7.44-3	$$⊢\left(B\in \mathrm{dom}\mathrm{rec}\left(F,A\right)\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup \mathrm{rec}\left(F,A\right)\left[B\right]$$