This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Definition df-vtx

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)

Ref Expression
Assertion df-vtx Vtx = ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cvtx ⊢ Vtx
1 vg ⊢ 𝑔
2 cvv ⊢ V
3 1 cv ⊢ 𝑔
4 2 2 cxp ⊢ ( V × V )
5 3 4 wcel ⊢ 𝑔 ∈ ( V × V )
6 c1st ⊢ 1st
7 3 6 cfv ⊢ ( 1st ‘ 𝑔 )
8 cbs ⊢ Base
9 3 8 cfv ⊢ ( Base ‘ 𝑔 )
10 5 7 9 cif ⊢ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) )
11 1 2 10 cmpt ⊢ ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) ) )
12 0 11 wceq ⊢ Vtx = ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) ) )