Theorem joinfval2

Description: Value of join function for a poset-type structure. (Contributed by NM, 12-Sep-2011) (Revised by NM, 9-Sep-2018)

	Ref	Expression
Hypotheses	joinfval.u	$$⊢U=\mathrm{lub}\left(K\right)$$
	joinfval.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
Assertion	joinfval2	$$⊢K\in V\to \underset{\dot{}}{\vee }=\left\{⟨⟨x,y⟩,z⟩|\left(\left\{x,y\right\}\in \mathrm{dom}U\wedge z=U\left(\left\{x,y\right\}\right)\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		joinfval.u	$$⊢U=\mathrm{lub}\left(K\right)$$
2		joinfval.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3	1 2	joinfval	$$⊢K\in V\to \underset{\dot{}}{\vee }=\left\{⟨⟨x,y⟩,z⟩|\left\{x,y\right\}Uz\right\}$$
4	1	lubfun	$$⊢\mathrm{Fun}U$$
5		funbrfv2b	$$⊢\mathrm{Fun}U\to \left(\left\{x,y\right\}Uz\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}U\wedge U\left(\left\{x,y\right\}\right)=z\right)\right)$$
6	4 5	ax-mp	$$⊢\left\{x,y\right\}Uz\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}U\wedge U\left(\left\{x,y\right\}\right)=z\right)$$
7		eqcom	$$⊢U\left(\left\{x,y\right\}\right)=z\leftrightarrow z=U\left(\left\{x,y\right\}\right)$$
8	7	anbi2i	$$⊢\left(\left\{x,y\right\}\in \mathrm{dom}U\wedge U\left(\left\{x,y\right\}\right)=z\right)\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}U\wedge z=U\left(\left\{x,y\right\}\right)\right)$$
9	6 8	bitri	$$⊢\left\{x,y\right\}Uz\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}U\wedge z=U\left(\left\{x,y\right\}\right)\right)$$
10	9	oprabbii	$$⊢\left\{⟨⟨x,y⟩,z⟩|\left\{x,y\right\}Uz\right\}=\left\{⟨⟨x,y⟩,z⟩|\left(\left\{x,y\right\}\in \mathrm{dom}U\wedge z=U\left(\left\{x,y\right\}\right)\right)\right\}$$
11	3 10	eqtrdi	$$⊢K\in V\to \underset{\dot{}}{\vee }=\left\{⟨⟨x,y⟩,z⟩|\left(\left\{x,y\right\}\in \mathrm{dom}U\wedge z=U\left(\left\{x,y\right\}\right)\right)\right\}$$