Definition df-vtx 28925

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 28923	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3447	. . 3 class V
4	2	cv 1539	. . . . 5 class 𝑔
5	3, 3	cxp 5636	. . . . 5 class (V × V)
6	4, 5	wcel 2109	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7966	. . . . 5 class 1st
8	4, 7	cfv 6511	. . . 4 class (1st ‘𝑔)
9		cbs 17179	. . . . 5 class Base
10	4, 9	cfv 6511	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4488	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5188	. 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  28927