Definition df-vtx 28824

Description: Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)

Assertion

Ref	Expression
df-vtx	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

Detailed syntax breakdown of Definition df-vtx

Step	Hyp	Ref	Expression
1		cvtx 28822	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3471	. . 3 class V
4	2	cv 1533	. . . . 5 class 𝑔
5	3, 3	cxp 5676	. . . . 5 class (V × V)
6	4, 5	wcel 2099	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7991	. . . . 5 class 1st
8	4, 7	cfv 6548	. . . 4 class (1st ‘𝑔)
9		cbs 17180	. . . . 5 class Base
10	4, 9	cfv 6548	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4529	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5231	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1534	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  28826