Definition df-vtx 29015

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 29013	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3480	. . 3 class V
4	2	cv 1539	. . . . 5 class 𝑔
5	3, 3	cxp 5683	. . . . 5 class (V × V)
6	4, 5	wcel 2108	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 8012	. . . . 5 class 1st
8	4, 7	cfv 6561	. . . 4 class (1st ‘𝑔)
9		cbs 17247	. . . . 5 class Base
10	4, 9	cfv 6561	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4525	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5225	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1540	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29017