Definition df-vtx 15938

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 15936	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2803	. . 3 class V
4	2	cv 1397	. . . . 5 class 𝑔
5	3, 3	cxp 4729	. . . . 5 class (V × V)
6	4, 5	wcel 2202	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6310	. . . . 5 class 1st
8	4, 7	cfv 5333	. . . 4 class (1st ‘𝑔)
9		cbs 13145	. . . . 5 class Base
10	4, 9	cfv 5333	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3607	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4155	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1398	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15940  1vgrex  15944  wlkreslem  16302  clwwlknonmpo  16352  trlsegvdegfi  16391  eupth2lem3lem1fi  16392  eupth2lem3lem2fi  16393  eupth2lem3lem6fi  16395  eupth2lem3lem4fi  16397