Theorem eliunov2uz

Description: Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. The index set N is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020)

	Ref	Expression
Hypothesis	eliunov2uz.def	$$⊢C=\left(r\in \mathrm{V}⟼\bigcup _{n\in N}\left(r\underset{\dot{}}{\times }n\right)\right)$$
Assertion	eliunov2uz	$$⊢\left(R\in U\wedge N={\mathbb{Z}}_{\ge M}\right)\to \left(X\in C\left(R\right)\leftrightarrow \exists n\in NX\in \left(R\underset{\dot{}}{\times }n\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eliunov2uz.def	$$⊢C=\left(r\in \mathrm{V}⟼\bigcup _{n\in N}\left(r\underset{\dot{}}{\times }n\right)\right)$$
2		simpr	$$⊢\left(R\in U\wedge N={\mathbb{Z}}_{\ge M}\right)\to N={\mathbb{Z}}_{\ge M}$$
3		fvex	$$⊢{\mathbb{Z}}_{\ge M}\in \mathrm{V}$$
4	2 3	eqeltrdi	$$⊢\left(R\in U\wedge N={\mathbb{Z}}_{\ge M}\right)\to N\in \mathrm{V}$$
5	1	eliunov2	$$⊢\left(R\in U\wedge N\in \mathrm{V}\right)\to \left(X\in C\left(R\right)\leftrightarrow \exists n\in NX\in \left(R\underset{\dot{}}{\times }n\right)\right)$$
6	4 5	syldan	$$⊢\left(R\in U\wedge N={\mathbb{Z}}_{\ge M}\right)\to \left(X\in C\left(R\right)\leftrightarrow \exists n\in NX\in \left(R\underset{\dot{}}{\times }n\right)\right)$$