Definition df-vtx 29145

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 29143	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3453	. . 3 class V
4	2	cv 1558	. . . . 5 class 𝑔
5	3, 3	cxp 5643	. . . . 5 class (V × V)
6	4, 5	wcel 2141	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7964	. . . . 5 class 1st
8	4, 7	cfv 6517	. . . 4 class (1st ‘𝑔)
9		cbs 17228	. . . . 5 class Base
10	4, 9	cfv 6517	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4479	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5180	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1559	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29147