Theorem rdglim

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

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

Proof

Step	Hyp	Ref	Expression
1		limelon	$$⊢\left(B\in C\wedge \mathrm{Lim}B\right)\to B\in \mathrm{On}$$
2		rdgfnon	$$⊢\mathrm{rec}\left(F,A\right)Fn\mathrm{On}$$
3		fndm	$$⊢\mathrm{rec}\left(F,A\right)Fn\mathrm{On}\to \mathrm{dom}\mathrm{rec}\left(F,A\right)=\mathrm{On}$$
4	2 3	ax-mp	$$⊢\mathrm{dom}\mathrm{rec}\left(F,A\right)=\mathrm{On}$$
5	1 4	eleqtrrdi	$$⊢\left(B\in C\wedge \mathrm{Lim}B\right)\to B\in \mathrm{dom}\mathrm{rec}\left(F,A\right)$$
6		rdglimg	$$⊢\left(B\in \mathrm{dom}\mathrm{rec}\left(F,A\right)\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup \mathrm{rec}\left(F,A\right)\left[B\right]$$
7	5 6	sylancom	$$⊢\left(B\in C\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup \mathrm{rec}\left(F,A\right)\left[B\right]$$