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Metamath Proof Explorer

Theorem usgredg2vtxeuALT

Description: Alternate proof of usgredg2vtxeu , using edgiedgb , the general translation from ( iEdgG ) to ( EdgG ) . (Contributed by AV, 18-Oct-2020) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion usgredg2vtxeuALT 
|- ( ( G e. USGraph /\ E e. ( Edg ` G ) /\ Y e. E ) -> E! y e. ( Vtx ` G ) E = { Y , y } )

Proof

Step Hyp Ref Expression
1 usgruhgr 
 |-  ( G e. USGraph -> G e. UHGraph )
2 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
3 2 uhgredgiedgb 
 |-  ( G e. UHGraph -> ( E e. ( Edg ` G ) <-> E. x e. dom ( iEdg ` G ) E = ( ( iEdg ` G ) ` x ) ) )
4 1 3 syl 
 |-  ( G e. USGraph -> ( E e. ( Edg ` G ) <-> E. x e. dom ( iEdg ` G ) E = ( ( iEdg ` G ) ` x ) ) )
5 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
6 5 2 usgredgreu 
 |-  ( ( G e. USGraph /\ x e. dom ( iEdg ` G ) /\ Y e. ( ( iEdg ` G ) ` x ) ) -> E! y e. ( Vtx ` G ) ( ( iEdg ` G ) ` x ) = { Y , y } )
7 6 3expia 
 |-  ( ( G e. USGraph /\ x e. dom ( iEdg ` G ) ) -> ( Y e. ( ( iEdg ` G ) ` x ) -> E! y e. ( Vtx ` G ) ( ( iEdg ` G ) ` x ) = { Y , y } ) )
8 7 3adant3 
 |-  ( ( G e. USGraph /\ x e. dom ( iEdg ` G ) /\ E = ( ( iEdg ` G ) ` x ) ) -> ( Y e. ( ( iEdg ` G ) ` x ) -> E! y e. ( Vtx ` G ) ( ( iEdg ` G ) ` x ) = { Y , y } ) )
9 eleq2 
 |-  ( E = ( ( iEdg ` G ) ` x ) -> ( Y e. E <-> Y e. ( ( iEdg ` G ) ` x ) ) )
10 eqeq1 
 |-  ( E = ( ( iEdg ` G ) ` x ) -> ( E = { Y , y } <-> ( ( iEdg ` G ) ` x ) = { Y , y } ) )
11 10 reubidv 
 |-  ( E = ( ( iEdg ` G ) ` x ) -> ( E! y e. ( Vtx ` G ) E = { Y , y } <-> E! y e. ( Vtx ` G ) ( ( iEdg ` G ) ` x ) = { Y , y } ) )
12 9 11 imbi12d 
 |-  ( E = ( ( iEdg ` G ) ` x ) -> ( ( Y e. E -> E! y e. ( Vtx ` G ) E = { Y , y } ) <-> ( Y e. ( ( iEdg ` G ) ` x ) -> E! y e. ( Vtx ` G ) ( ( iEdg ` G ) ` x ) = { Y , y } ) ) )
13 12 3ad2ant3 
 |-  ( ( G e. USGraph /\ x e. dom ( iEdg ` G ) /\ E = ( ( iEdg ` G ) ` x ) ) -> ( ( Y e. E -> E! y e. ( Vtx ` G ) E = { Y , y } ) <-> ( Y e. ( ( iEdg ` G ) ` x ) -> E! y e. ( Vtx ` G ) ( ( iEdg ` G ) ` x ) = { Y , y } ) ) )
14 8 13 mpbird 
 |-  ( ( G e. USGraph /\ x e. dom ( iEdg ` G ) /\ E = ( ( iEdg ` G ) ` x ) ) -> ( Y e. E -> E! y e. ( Vtx ` G ) E = { Y , y } ) )
15 14 3exp 
 |-  ( G e. USGraph -> ( x e. dom ( iEdg ` G ) -> ( E = ( ( iEdg ` G ) ` x ) -> ( Y e. E -> E! y e. ( Vtx ` G ) E = { Y , y } ) ) ) )
16 15 rexlimdv 
 |-  ( G e. USGraph -> ( E. x e. dom ( iEdg ` G ) E = ( ( iEdg ` G ) ` x ) -> ( Y e. E -> E! y e. ( Vtx ` G ) E = { Y , y } ) ) )
17 4 16 sylbid 
 |-  ( G e. USGraph -> ( E e. ( Edg ` G ) -> ( Y e. E -> E! y e. ( Vtx ` G ) E = { Y , y } ) ) )
18 17 3imp 
 |-  ( ( G e. USGraph /\ E e. ( Edg ` G ) /\ Y e. E ) -> E! y e. ( Vtx ` G ) E = { Y , y } )