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Metamath Proof Explorer

Theorem vtxdumgrval

Description: The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017) (Revised by AV, 23-Feb-2021)

Ref Expression
Hypotheses vtxdlfgrval.v 
|- V = ( Vtx ` G )
vtxdlfgrval.i 
|- I = ( iEdg ` G )
vtxdlfgrval.a 
|- A = dom I
vtxdlfgrval.d 
|- D = ( VtxDeg ` G )
Assertion vtxdumgrval 
|- ( ( G e. UMGraph /\ U e. V ) -> ( D ` U ) = ( # ` { x e. A | U e. ( I ` x ) } ) )

Proof

Step Hyp Ref Expression
1 vtxdlfgrval.v 
 |-  V = ( Vtx ` G )
2 vtxdlfgrval.i 
 |-  I = ( iEdg ` G )
3 vtxdlfgrval.a 
 |-  A = dom I
4 vtxdlfgrval.d 
 |-  D = ( VtxDeg ` G )
5 1 2 umgrislfupgr 
 |-  ( G e. UMGraph <-> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) )
6 3 eqcomi 
 |-  dom I = A
7 6 feq2i 
 |-  ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> I : A --> { x e. ~P V | 2 <_ ( # ` x ) } )
8 7 biimpi 
 |-  ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> I : A --> { x e. ~P V | 2 <_ ( # ` x ) } )
9 5 8 simplbiim 
 |-  ( G e. UMGraph -> I : A --> { x e. ~P V | 2 <_ ( # ` x ) } )
10 1 2 3 4 vtxdlfgrval 
 |-  ( ( I : A --> { x e. ~P V | 2 <_ ( # ` x ) } /\ U e. V ) -> ( D ` U ) = ( # ` { x e. A | U e. ( I ` x ) } ) )
11 9 10 sylan 
 |-  ( ( G e. UMGraph /\ U e. V ) -> ( D ` U ) = ( # ` { x e. A | U e. ( I ` x ) } ) )