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Theorem 2pthon3v-av 40065
Description: For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)
Hypotheses
Ref Expression
2pthon3v-av.v  |-  V  =  (Vtx `  G )
2pthon3v-av.e  |-  E  =  (Edg `  G )
Assertion
Ref Expression
2pthon3v-av  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C )  /\  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E
) )  ->  E. f E. p ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 ) )
Distinct variable groups:    A, f, p    B, f, p    C, f, p    f, G, p
Allowed substitution hints:    E( f, p)    V( f, p)

Proof of Theorem 2pthon3v-av
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2pthon3v-av.e . . . . . . . . . 10  |-  E  =  (Edg `  G )
2 edgaval 39373 . . . . . . . . . 10  |-  ( G  e. UHGraph  ->  (Edg `  G
)  =  ran  (iEdg `  G ) )
31, 2syl5eq 2517 . . . . . . . . 9  |-  ( G  e. UHGraph  ->  E  =  ran  (iEdg `  G ) )
43eleq2d 2534 . . . . . . . 8  |-  ( G  e. UHGraph  ->  ( { A ,  B }  e.  E  <->  { A ,  B }  e.  ran  (iEdg `  G
) ) )
5 2pthon3v-av.v . . . . . . . . . . 11  |-  V  =  (Vtx `  G )
6 eqid 2471 . . . . . . . . . . 11  |-  (iEdg `  G )  =  (iEdg `  G )
75, 6uhgrf 39306 . . . . . . . . . 10  |-  ( G  e. UHGraph  ->  (iEdg `  G
) : dom  (iEdg `  G ) --> ( ~P V  \  { (/) } ) )
87ffnd 5740 . . . . . . . . 9  |-  ( G  e. UHGraph  ->  (iEdg `  G
)  Fn  dom  (iEdg `  G ) )
9 fvelrnb 5926 . . . . . . . . 9  |-  ( (iEdg `  G )  Fn  dom  (iEdg `  G )  -> 
( { A ,  B }  e.  ran  (iEdg `  G )  <->  E. i  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  i
)  =  { A ,  B } ) )
108, 9syl 17 . . . . . . . 8  |-  ( G  e. UHGraph  ->  ( { A ,  B }  e.  ran  (iEdg `  G )  <->  E. i  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  i
)  =  { A ,  B } ) )
114, 10bitrd 261 . . . . . . 7  |-  ( G  e. UHGraph  ->  ( { A ,  B }  e.  E  <->  E. i  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  i
)  =  { A ,  B } ) )
123eleq2d 2534 . . . . . . . 8  |-  ( G  e. UHGraph  ->  ( { B ,  C }  e.  E  <->  { B ,  C }  e.  ran  (iEdg `  G
) ) )
13 fvelrnb 5926 . . . . . . . . 9  |-  ( (iEdg `  G )  Fn  dom  (iEdg `  G )  -> 
( { B ,  C }  e.  ran  (iEdg `  G )  <->  E. j  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  j
)  =  { B ,  C } ) )
148, 13syl 17 . . . . . . . 8  |-  ( G  e. UHGraph  ->  ( { B ,  C }  e.  ran  (iEdg `  G )  <->  E. j  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  j
)  =  { B ,  C } ) )
1512, 14bitrd 261 . . . . . . 7  |-  ( G  e. UHGraph  ->  ( { B ,  C }  e.  E  <->  E. j  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  j
)  =  { B ,  C } ) )
1611, 15anbi12d 725 . . . . . 6  |-  ( G  e. UHGraph  ->  ( ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E
)  <->  ( E. i  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  E. j  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  j
)  =  { B ,  C } ) ) )
1716adantr 472 . . . . 5  |-  ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E
)  <->  ( E. i  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  E. j  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  j
)  =  { B ,  C } ) ) )
1817adantr 472 . . . 4  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  <->  ( E. i  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  i )  =  { A ,  B }  /\  E. j  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  j )  =  { B ,  C }
) ) )
19 reeanv 2944 . . . 4  |-  ( E. i  e.  dom  (iEdg `  G ) E. j  e.  dom  (iEdg `  G
) ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
)  <->  ( E. i  e.  dom  (iEdg `  G
) ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  E. j  e.  dom  (iEdg `  G ) ( (iEdg `  G ) `  j
)  =  { B ,  C } ) )
2018, 19syl6bbr 271 . . 3  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  <->  E. i  e.  dom  (iEdg `  G ) E. j  e.  dom  (iEdg `  G
) ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) ) )
21 df-s2 13003 . . . . . . . 8  |-  <" i
j ">  =  ( <" i "> ++  <" j "> )
22 ovex 6336 . . . . . . . 8  |-  ( <" i "> ++  <" j "> )  e.  _V
2321, 22eqeltri 2545 . . . . . . 7  |-  <" i
j ">  e.  _V
24 df-s3 13004 . . . . . . . 8  |-  <" A B C ">  =  ( <" A B "> ++  <" C "> )
25 ovex 6336 . . . . . . . 8  |-  ( <" A B "> ++  <" C "> )  e.  _V
2624, 25eqeltri 2545 . . . . . . 7  |-  <" A B C ">  e.  _V
2723, 26pm3.2i 462 . . . . . 6  |-  ( <" i j ">  e.  _V  /\  <" A B C ">  e.  _V )
28 eqid 2471 . . . . . . . 8  |-  <" A B C ">  =  <" A B C ">
29 eqid 2471 . . . . . . . 8  |-  <" i
j ">  =  <" i j ">
30 simp-4r 785 . . . . . . . 8  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)
31 3simpb 1028 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( A  =/=  B  /\  B  =/=  C ) )
3231ad3antlr 745 . . . . . . . 8  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  ( A  =/=  B  /\  B  =/=  C ) )
33 eqimss2 3471 . . . . . . . . . 10  |-  ( ( (iEdg `  G ) `  i )  =  { A ,  B }  ->  { A ,  B }  C_  ( (iEdg `  G ) `  i
) )
34 eqimss2 3471 . . . . . . . . . 10  |-  ( ( (iEdg `  G ) `  j )  =  { B ,  C }  ->  { B ,  C }  C_  ( (iEdg `  G ) `  j
) )
3533, 34anim12i 576 . . . . . . . . 9  |-  ( ( ( (iEdg `  G
) `  i )  =  { A ,  B }  /\  ( (iEdg `  G ) `  j
)  =  { B ,  C } )  -> 
( { A ,  B }  C_  ( (iEdg `  G ) `  i
)  /\  { B ,  C }  C_  (
(iEdg `  G ) `  j ) ) )
3635adantl 473 . . . . . . . 8  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  ( { A ,  B }  C_  ( (iEdg `  G
) `  i )  /\  { B ,  C }  C_  ( (iEdg `  G ) `  j
) ) )
37 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( i  =  j  ->  (
(iEdg `  G ) `  i )  =  ( (iEdg `  G ) `  j ) )
3837eqeq1d 2473 . . . . . . . . . . . . . 14  |-  ( i  =  j  ->  (
( (iEdg `  G
) `  i )  =  { A ,  B } 
<->  ( (iEdg `  G
) `  j )  =  { A ,  B } ) )
3938anbi1d 719 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
)  <->  ( ( (iEdg `  G ) `  j
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) ) )
40 eqtr2 2491 . . . . . . . . . . . . . 14  |-  ( ( ( (iEdg `  G
) `  j )  =  { A ,  B }  /\  ( (iEdg `  G ) `  j
)  =  { B ,  C } )  ->  { A ,  B }  =  { B ,  C } )
41 3simpa 1027 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  B  e.  V
) )
42 3simpc 1029 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( B  e.  V  /\  C  e.  V
) )
43 preq12bg 4146 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( { A ,  B }  =  { B ,  C }  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) ) )
4441, 42, 43syl2anc 673 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( { A ,  B }  =  { B ,  C }  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) ) )
45 eqneqall 2654 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A  =  B  ->  ( A  =/=  B  ->  i  =/=  j ) )
4645com12 31 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  B  ->  ( A  =  B  ->  i  =/=  j ) )
47463ad2ant1 1051 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( A  =  B  ->  i  =/=  j ) )
4847com12 31 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  =  B  ->  (
( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
)  ->  i  =/=  j ) )
4948adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  =  B  /\  B  =  C )  ->  ( ( A  =/= 
B  /\  A  =/=  C  /\  B  =/=  C
)  ->  i  =/=  j ) )
50 eqneqall 2654 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A  =  C  ->  ( A  =/=  C  ->  i  =/=  j ) )
5150com12 31 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  C  ->  ( A  =  C  ->  i  =/=  j ) )
52513ad2ant2 1052 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  ( A  =  C  ->  i  =/=  j ) )
5352com12 31 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  =  C  ->  (
( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
)  ->  i  =/=  j ) )
5453adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  =  C  /\  B  =  B )  ->  ( ( A  =/= 
B  /\  A  =/=  C  /\  B  =/=  C
)  ->  i  =/=  j ) )
5549, 54jaoi 386 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) )  -> 
( ( A  =/= 
B  /\  A  =/=  C  /\  B  =/=  C
)  ->  i  =/=  j ) )
5644, 55syl6bi 236 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( { A ,  B }  =  { B ,  C }  ->  ( ( A  =/= 
B  /\  A  =/=  C  /\  B  =/=  C
)  ->  i  =/=  j ) ) )
5756com23 80 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( A  =/= 
B  /\  A  =/=  C  /\  B  =/=  C
)  ->  ( { A ,  B }  =  { B ,  C }  ->  i  =/=  j
) ) )
5857adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C )  ->  ( { A ,  B }  =  { B ,  C }  ->  i  =/=  j
) ) )
5958imp 436 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  ( { A ,  B }  =  { B ,  C }  ->  i  =/=  j
) )
6059com12 31 . . . . . . . . . . . . . 14  |-  ( { A ,  B }  =  { B ,  C }  ->  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  i  =/=  j ) )
6140, 60syl 17 . . . . . . . . . . . . 13  |-  ( ( ( (iEdg `  G
) `  j )  =  { A ,  B }  /\  ( (iEdg `  G ) `  j
)  =  { B ,  C } )  -> 
( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  ->  i  =/=  j ) )
6239, 61syl6bi 236 . . . . . . . . . . . 12  |-  ( i  =  j  ->  (
( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
)  ->  ( (
( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  i  =/=  j ) ) )
6362com23 80 . . . . . . . . . . 11  |-  ( i  =  j  ->  (
( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
)  ->  i  =/=  j ) ) )
64 2a1 27 . . . . . . . . . . 11  |-  ( i  =/=  j  ->  (
( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
)  ->  i  =/=  j ) ) )
6563, 64pm2.61ine 2726 . . . . . . . . . 10  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
)  ->  i  =/=  j ) )
6665adantr 472 . . . . . . . . 9  |-  ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  ( i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G ) ) )  ->  ( ( ( (iEdg `  G ) `  i )  =  { A ,  B }  /\  ( (iEdg `  G
) `  j )  =  { B ,  C } )  ->  i  =/=  j ) )
6766imp 436 . . . . . . . 8  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  i  =/=  j )
68 simplr2 1073 . . . . . . . . 9  |-  ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  ( i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G ) ) )  ->  A  =/=  C
)
6968adantr 472 . . . . . . . 8  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  A  =/=  C )
7028, 29, 30, 32, 36, 5, 6, 67, 692pthond 40064 . . . . . . 7  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  <" i
j "> ( A (SPathsOn `  G ) C ) <" A B C "> )
71 s2len 13043 . . . . . . 7  |-  ( # `  <" i j "> )  =  2
7270, 71jctir 547 . . . . . 6  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  ( <" i j "> ( A (SPathsOn `  G ) C )
<" A B C ">  /\  ( # `
 <" i j "> )  =  2 ) )
73 breq12 4400 . . . . . . . 8  |-  ( ( f  =  <" i
j ">  /\  p  =  <" A B C "> )  ->  ( f ( A (SPathsOn `  G ) C ) p  <->  <" i
j "> ( A (SPathsOn `  G ) C ) <" A B C "> )
)
74 fveq2 5879 . . . . . . . . . 10  |-  ( f  =  <" i j ">  ->  ( # `
 f )  =  ( # `  <" i j "> ) )
7574eqeq1d 2473 . . . . . . . . 9  |-  ( f  =  <" i j ">  ->  (
( # `  f )  =  2  <->  ( # `  <" i j "> )  =  2 ) )
7675adantr 472 . . . . . . . 8  |-  ( ( f  =  <" i
j ">  /\  p  =  <" A B C "> )  ->  ( ( # `  f
)  =  2  <->  ( # `
 <" i j "> )  =  2 ) )
7773, 76anbi12d 725 . . . . . . 7  |-  ( ( f  =  <" i
j ">  /\  p  =  <" A B C "> )  ->  ( ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 )  <-> 
( <" i j "> ( A (SPathsOn `  G ) C ) <" A B C ">  /\  ( # `
 <" i j "> )  =  2 ) ) )
7877spc2egv 3122 . . . . . 6  |-  ( (
<" i j ">  e.  _V  /\  <" A B C ">  e.  _V )  ->  ( ( <" i j "> ( A (SPathsOn `  G ) C )
<" A B C ">  /\  ( # `
 <" i j "> )  =  2 )  ->  E. f E. p ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 ) ) )
7927, 72, 78mpsyl 64 . . . . 5  |-  ( ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G
) ) )  /\  ( ( (iEdg `  G ) `  i
)  =  { A ,  B }  /\  (
(iEdg `  G ) `  j )  =  { B ,  C }
) )  ->  E. f E. p ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 ) )
8079ex 441 . . . 4  |-  ( ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  ( i  e.  dom  (iEdg `  G )  /\  j  e.  dom  (iEdg `  G ) ) )  ->  ( ( ( (iEdg `  G ) `  i )  =  { A ,  B }  /\  ( (iEdg `  G
) `  j )  =  { B ,  C } )  ->  E. f E. p ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 ) ) )
8180rexlimdvva 2878 . . 3  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  ( E. i  e.  dom  (iEdg `  G ) E. j  e.  dom  (iEdg `  G ) ( ( (iEdg `  G ) `  i )  =  { A ,  B }  /\  ( (iEdg `  G
) `  j )  =  { B ,  C } )  ->  E. f E. p ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 ) ) )
8220, 81sylbid 223 . 2  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  ->  E. f E. p
( f ( A (SPathsOn `  G ) C ) p  /\  ( # `  f )  =  2 ) ) )
83823impia 1228 1  |-  ( ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C )  /\  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E
) )  ->  E. f E. p ( f ( A (SPathsOn `  G
) C ) p  /\  ( # `  f
)  =  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   {cpr 3961   class class class wbr 4395   dom cdm 4839   ran crn 4840    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   2c2 10681   #chash 12553   ++ cconcat 12705   <"cs1 12706   <"cs2 12996   <"cs3 12997  Vtxcvtx 39251  iEdgciedg 39252   UHGraph cuhgr 39300  Edgcedga 39371  SPathsOncspthson 39910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-uhgr 39302  df-edga 39372  df-1wlks 39804  df-wlkson 39806  df-trls 39889  df-trlson 39890  df-pths 39911  df-spths 39912  df-spthson 39914
This theorem is referenced by: (None)
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