Definition df-vtx 29067

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 29065	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3429	. . 3 class V
4	2	cv 1541	. . . . 5 class 𝑔
5	3, 3	cxp 5629	. . . . 5 class (V × V)
6	4, 5	wcel 2114	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7940	. . . . 5 class 1st
8	4, 7	cfv 6498	. . . 4 class (1st ‘𝑔)
9		cbs 17179	. . . . 5 class Base
10	4, 9	cfv 6498	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4466	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5166	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1542	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29069