This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem usgredgprvALT

Description: Alternate proof of usgredgprv , using usgredg2 instead of umgredgprv . (Contributed by Alexander van der Vekens, 19-Aug-2017) (Revised by AV, 16-Oct-2020) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses usgredg2.e 
|- E = ( iEdg ` G )
usgredgprv.v 
|- V = ( Vtx ` G )
Assertion usgredgprvALT 
|- ( ( G e. USGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )

Proof

Step Hyp Ref Expression
1 usgredg2.e 
 |-  E = ( iEdg ` G )
2 usgredgprv.v 
 |-  V = ( Vtx ` G )
3 1 2 usgrss 
 |-  ( ( G e. USGraph /\ X e. dom E ) -> ( E ` X ) C_ V )
4 1 usgredg2 
 |-  ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )
5 sseq1 
 |-  ( ( E ` X ) = { M , N } -> ( ( E ` X ) C_ V <-> { M , N } C_ V ) )
6 fveq2 
 |-  ( ( E ` X ) = { M , N } -> ( # ` ( E ` X ) ) = ( # ` { M , N } ) )
7 6 eqeq1d 
 |-  ( ( E ` X ) = { M , N } -> ( ( # ` ( E ` X ) ) = 2 <-> ( # ` { M , N } ) = 2 ) )
8 5 7 anbi12d 
 |-  ( ( E ` X ) = { M , N } -> ( ( ( E ` X ) C_ V /\ ( # ` ( E ` X ) ) = 2 ) <-> ( { M , N } C_ V /\ ( # ` { M , N } ) = 2 ) ) )
9 eqid 
 |-  { M , N } = { M , N }
10 9 hashprdifel 
 |-  ( ( # ` { M , N } ) = 2 -> ( M e. { M , N } /\ N e. { M , N } /\ M =/= N ) )
11 prssg 
 |-  ( ( M e. { M , N } /\ N e. { M , N } ) -> ( ( M e. V /\ N e. V ) <-> { M , N } C_ V ) )
12 11 3adant3 
 |-  ( ( M e. { M , N } /\ N e. { M , N } /\ M =/= N ) -> ( ( M e. V /\ N e. V ) <-> { M , N } C_ V ) )
13 12 biimprd 
 |-  ( ( M e. { M , N } /\ N e. { M , N } /\ M =/= N ) -> ( { M , N } C_ V -> ( M e. V /\ N e. V ) ) )
14 10 13 syl 
 |-  ( ( # ` { M , N } ) = 2 -> ( { M , N } C_ V -> ( M e. V /\ N e. V ) ) )
15 14 impcom 
 |-  ( ( { M , N } C_ V /\ ( # ` { M , N } ) = 2 ) -> ( M e. V /\ N e. V ) )
16 8 15 biimtrdi 
 |-  ( ( E ` X ) = { M , N } -> ( ( ( E ` X ) C_ V /\ ( # ` ( E ` X ) ) = 2 ) -> ( M e. V /\ N e. V ) ) )
17 16 com12 
 |-  ( ( ( E ` X ) C_ V /\ ( # ` ( E ` X ) ) = 2 ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )
18 3 4 17 syl2anc 
 |-  ( ( G e. USGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )