Definition df-vtx 40212

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 40210	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3172	. . 3 class V
4	2	cv 1473	. . . . 5 class 𝑔
5	3, 3	cxp 5025	. . . . 5 class (V × V)
6	4, 5	wcel 1976	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7034	. . . . 5 class 1st
8	4, 7	cfv 5789	. . . 4 class (1st ‘𝑔)
9		cbs 15643	. . . . 5 class Base
10	4, 9	cfv 5789	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4035	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4637	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1474	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  40214