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Metamath Proof Explorer

Theorem upgredg2vtx

Description: For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020) (Revised by AV, 5-Dec-2020)

Ref Expression
Hypotheses upgredg.v 
|- V = ( Vtx ` G )
upgredg.e 
|- E = ( Edg ` G )
Assertion upgredg2vtx 
|- ( ( G e. UPGraph /\ C e. E /\ A e. C ) -> E. b e. V C = { A , b } )

Proof

Step Hyp Ref Expression
1 upgredg.v 
 |-  V = ( Vtx ` G )
2 upgredg.e 
 |-  E = ( Edg ` G )
3 1 2 upgredg 
 |-  ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. c e. V C = { a , c } )
4 3 3adant3 
 |-  ( ( G e. UPGraph /\ C e. E /\ A e. C ) -> E. a e. V E. c e. V C = { a , c } )
5 elpr2elpr 
 |-  ( ( a e. V /\ c e. V /\ A e. { a , c } ) -> E. b e. V { a , c } = { A , b } )
6 5 3expia 
 |-  ( ( a e. V /\ c e. V ) -> ( A e. { a , c } -> E. b e. V { a , c } = { A , b } ) )
7 eleq2 
 |-  ( C = { a , c } -> ( A e. C <-> A e. { a , c } ) )
8 eqeq1 
 |-  ( C = { a , c } -> ( C = { A , b } <-> { a , c } = { A , b } ) )
9 8 rexbidv 
 |-  ( C = { a , c } -> ( E. b e. V C = { A , b } <-> E. b e. V { a , c } = { A , b } ) )
10 7 9 imbi12d 
 |-  ( C = { a , c } -> ( ( A e. C -> E. b e. V C = { A , b } ) <-> ( A e. { a , c } -> E. b e. V { a , c } = { A , b } ) ) )
11 6 10 imbitrrid 
 |-  ( C = { a , c } -> ( ( a e. V /\ c e. V ) -> ( A e. C -> E. b e. V C = { A , b } ) ) )
12 11 com13 
 |-  ( A e. C -> ( ( a e. V /\ c e. V ) -> ( C = { a , c } -> E. b e. V C = { A , b } ) ) )
13 12 3ad2ant3 
 |-  ( ( G e. UPGraph /\ C e. E /\ A e. C ) -> ( ( a e. V /\ c e. V ) -> ( C = { a , c } -> E. b e. V C = { A , b } ) ) )
14 13 rexlimdvv 
 |-  ( ( G e. UPGraph /\ C e. E /\ A e. C ) -> ( E. a e. V E. c e. V C = { a , c } -> E. b e. V C = { A , b } ) )
15 4 14 mpd 
 |-  ( ( G e. UPGraph /\ C e. E /\ A e. C ) -> E. b e. V C = { A , b } )