Theorem fiuni

Description: The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009) (Revised by Mario Carneiro, 24-Nov-2013)

	Ref	Expression
Assertion	fiuni	$$⊢A\in V\to \bigcup A=\bigcup \mathrm{fi}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssfii	$$⊢A\in V\to A\subseteq \mathrm{fi}\left(A\right)$$
2	1	unissd	$$⊢A\in V\to \bigcup A\subseteq \bigcup \mathrm{fi}\left(A\right)$$
3		fipwuni	$$⊢\mathrm{fi}\left(A\right)\subseteq 𝒫\bigcup A$$
4	3	unissi	$$⊢\bigcup \mathrm{fi}\left(A\right)\subseteq \bigcup 𝒫\bigcup A$$
5		unipw	$$⊢\bigcup 𝒫\bigcup A=\bigcup A$$
6	4 5	sseqtri	$$⊢\bigcup \mathrm{fi}\left(A\right)\subseteq \bigcup A$$
7	6	a1i	$$⊢A\in V\to \bigcup \mathrm{fi}\left(A\right)\subseteq \bigcup A$$
8	2 7	eqssd	$$⊢A\in V\to \bigcup A=\bigcup \mathrm{fi}\left(A\right)$$