Definition df-vtx 25921

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 25919	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3231	. . 3 class V
4	2	cv 1522	. . . . 5 class 𝑔
5	3, 3	cxp 5141	. . . . 5 class (V × V)
6	4, 5	wcel 2030	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7208	. . . . 5 class 1st
8	4, 7	cfv 5926	. . . 4 class (1st ‘𝑔)
9		cbs 15904	. . . . 5 class Base
10	4, 9	cfv 5926	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4119	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4762	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1523	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  25923  vtxvalOLD  25925