Theorem climfv

Description: The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Assertion	climfv	$$⊢F⇝A\to A=⇝\left(F\right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢F⇝A\to F⇝A$$
2		climrel	$$⊢\mathrm{Rel}⇝$$
3	2	a1i	$$⊢F⇝A\to \mathrm{Rel}⇝$$
4		brrelex1	$$⊢\left(\mathrm{Rel}⇝\wedge F⇝A\right)\to F\in \mathrm{V}$$
5	3 1 4	syl2anc	$$⊢F⇝A\to F\in \mathrm{V}$$
6		brrelex2	$$⊢\left(\mathrm{Rel}⇝\wedge F⇝A\right)\to A\in \mathrm{V}$$
7	3 1 6	syl2anc	$$⊢F⇝A\to A\in \mathrm{V}$$
8		breldmg	$$⊢\left(F\in \mathrm{V}\wedge A\in \mathrm{V}\wedge F⇝A\right)\to F\in \mathrm{dom}⇝$$
9	5 7 1 8	syl3anc	$$⊢F⇝A\to F\in \mathrm{dom}⇝$$
10		climdm	$$⊢F\in \mathrm{dom}⇝\leftrightarrow F⇝⇝\left(F\right)$$
11	9 10	sylib	$$⊢F⇝A\to F⇝⇝\left(F\right)$$
12		climuni	$$⊢\left(F⇝A\wedge F⇝⇝\left(F\right)\right)\to A=⇝\left(F\right)$$
13	1 11 12	syl2anc	$$⊢F⇝A\to A=⇝\left(F\right)$$