Definition df-vtx 15855

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 15853	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2800	. . 3 class V
4	2	cv 1394	. . . . 5 class 𝑔
5	3, 3	cxp 4721	. . . . 5 class (V × V)
6	4, 5	wcel 2200	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6296	. . . . 5 class 1st
8	4, 7	cfv 5324	. . . 4 class (1st ‘𝑔)
9		cbs 13072	. . . . 5 class Base
10	4, 9	cfv 5324	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3603	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4148	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1395	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15857  1vgrex  15861  wlkreslem  16173  clwwlknonmpo  16223