Theorem rdglim2a

Description: The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998)

	Ref	Expression
Assertion	rdglim2a	$$⊢\left(B\in C\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup _{x\in B}\mathrm{rec}\left(F,A\right)\left(x\right)$$

Proof

Step	Hyp	Ref	Expression
1		rdglim2	$$⊢\left(B\in C\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup \left\{y|\exists x\in By=\mathrm{rec}\left(F,A\right)\left(x\right)\right\}$$
2		fvex	$$⊢\mathrm{rec}\left(F,A\right)\left(x\right)\in \mathrm{V}$$
3	2	dfiun2	$$⊢\bigcup _{x\in B}\mathrm{rec}\left(F,A\right)\left(x\right)=\bigcup \left\{y|\exists x\in By=\mathrm{rec}\left(F,A\right)\left(x\right)\right\}$$
4	1 3	eqtr4di	$$⊢\left(B\in C\wedge \mathrm{Lim}B\right)\to \mathrm{rec}\left(F,A\right)\left(B\right)=\bigcup _{x\in B}\mathrm{rec}\left(F,A\right)\left(x\right)$$