Definition df-vtx 28977

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 28975	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3459	. . 3 class V
4	2	cv 1539	. . . . 5 class 𝑔
5	3, 3	cxp 5652	. . . . 5 class (V × V)
6	4, 5	wcel 2108	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7986	. . . . 5 class 1st
8	4, 7	cfv 6531	. . . 4 class (1st ‘𝑔)
9		cbs 17228	. . . . 5 class Base
10	4, 9	cfv 6531	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4500	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5201	. 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  28979