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Metamath Proof Explorer

Theorem usgredg2ALT

Description: Alternate proof of usgredg2 , not using umgredg2 . (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 16-Oct-2020) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis usgredg2.e 
|- E = ( iEdg ` G )
Assertion usgredg2ALT 
|- ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )

Proof

Step Hyp Ref Expression
1 usgredg2.e 
 |-  E = ( iEdg ` G )
2 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
3 2 1 usgrf 
 |-  ( G e. USGraph -> E : dom E -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
4 f1f 
 |-  ( E : dom E -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> E : dom E --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
5 3 4 syl 
 |-  ( G e. USGraph -> E : dom E --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
6 5 ffvelcdmda 
 |-  ( ( G e. USGraph /\ X e. dom E ) -> ( E ` X ) e. { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
7 fveq2 
 |-  ( x = ( E ` X ) -> ( # ` x ) = ( # ` ( E ` X ) ) )
8 7 eqeq1d 
 |-  ( x = ( E ` X ) -> ( ( # ` x ) = 2 <-> ( # ` ( E ` X ) ) = 2 ) )
9 8 elrab 
 |-  ( ( E ` X ) e. { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } <-> ( ( E ` X ) e. ( ~P ( Vtx ` G ) \ { (/) } ) /\ ( # ` ( E ` X ) ) = 2 ) )
10 9 simprbi 
 |-  ( ( E ` X ) e. { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> ( # ` ( E ` X ) ) = 2 )
11 6 10 syl 
 |-  ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )