Theorem meetfval2

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

	Ref	Expression
Hypotheses	meetfval.u	$$⊢G=\mathrm{glb}\left(K\right)$$
	meetfval.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
Assertion	meetfval2	$$⊢K\in V\to \underset{\dot{}}{\wedge }=\left\{⟨⟨x,y⟩,z⟩|\left(\left\{x,y\right\}\in \mathrm{dom}G\wedge z=G\left(\left\{x,y\right\}\right)\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		meetfval.u	$$⊢G=\mathrm{glb}\left(K\right)$$
2		meetfval.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
3	1 2	meetfval	$$⊢K\in V\to \underset{\dot{}}{\wedge }=\left\{⟨⟨x,y⟩,z⟩|\left\{x,y\right\}Gz\right\}$$
4	1	glbfun	$$⊢\mathrm{Fun}G$$
5		funbrfv2b	$$⊢\mathrm{Fun}G\to \left(\left\{x,y\right\}Gz\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}G\wedge G\left(\left\{x,y\right\}\right)=z\right)\right)$$
6	4 5	ax-mp	$$⊢\left\{x,y\right\}Gz\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}G\wedge G\left(\left\{x,y\right\}\right)=z\right)$$
7		eqcom	$$⊢G\left(\left\{x,y\right\}\right)=z\leftrightarrow z=G\left(\left\{x,y\right\}\right)$$
8	7	anbi2i	$$⊢\left(\left\{x,y\right\}\in \mathrm{dom}G\wedge G\left(\left\{x,y\right\}\right)=z\right)\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}G\wedge z=G\left(\left\{x,y\right\}\right)\right)$$
9	6 8	bitri	$$⊢\left\{x,y\right\}Gz\leftrightarrow \left(\left\{x,y\right\}\in \mathrm{dom}G\wedge z=G\left(\left\{x,y\right\}\right)\right)$$
10	9	oprabbii	$$⊢\left\{⟨⟨x,y⟩,z⟩|\left\{x,y\right\}Gz\right\}=\left\{⟨⟨x,y⟩,z⟩|\left(\left\{x,y\right\}\in \mathrm{dom}G\wedge z=G\left(\left\{x,y\right\}\right)\right)\right\}$$
11	3 10	eqtrdi	$$⊢K\in V\to \underset{\dot{}}{\wedge }=\left\{⟨⟨x,y⟩,z⟩|\left(\left\{x,y\right\}\in \mathrm{dom}G\wedge z=G\left(\left\{x,y\right\}\right)\right)\right\}$$