Definition df-vtx 26096

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 26094	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3340	. . 3 class V
4	2	cv 1631	. . . . 5 class 𝑔
5	3, 3	cxp 5264	. . . . 5 class (V × V)
6	4, 5	wcel 2139	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7332	. . . . 5 class 1st
8	4, 7	cfv 6049	. . . 4 class (1st ‘𝑔)
9		cbs 16079	. . . . 5 class Base
10	4, 9	cfv 6049	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4230	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4881	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1632	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  26098  vtxvalOLD  26100