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Metamath Proof Explorer

Theorem upgrfn

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

Ref Expression
Hypotheses isupgr.v 
|- V = ( Vtx ` G )
isupgr.e 
|- E = ( iEdg ` G )
Assertion upgrfn 
|- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )

Proof

Step Hyp Ref Expression
1 isupgr.v 
 |-  V = ( Vtx ` G )
2 isupgr.e 
 |-  E = ( iEdg ` G )
3 1 2 upgrf 
 |-  ( G e. UPGraph -> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
4 fndm 
 |-  ( E Fn A -> dom E = A )
5 4 feq2d 
 |-  ( E Fn A -> ( E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
6 3 5 syl5ibcom 
 |-  ( G e. UPGraph -> ( E Fn A -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
7 6 imp 
 |-  ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )