Definition df-vtx 25876

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 25874	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3200	. . 3 class V
4	2	cv 1482	. . . . 5 class 𝑔
5	3, 3	cxp 5112	. . . . 5 class (V × V)
6	4, 5	wcel 1990	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7166	. . . . 5 class 1st
8	4, 7	cfv 5888	. . . 4 class (1st ‘𝑔)
9		cbs 15857	. . . . 5 class Base
10	4, 9	cfv 5888	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4086	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4729	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1483	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  25878  vtxvalOLD  25880