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Theorem 2pthfrgra 28338
Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Assertion
Ref Expression
2pthfrgra  |-  ( V FriendGrph  E  ->  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. f E. p ( f ( a ( V PathOn  E
) b ) p  /\  ( # `  f
)  =  2 ) )
Distinct variable groups:    V, a,
b, f, p    E, a, b, f, p

Proof of Theorem 2pthfrgra
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 2pthfrgrarn2 28337 . 2  |-  ( V FriendGrph  E  ->  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. m  e.  V  ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) ) )
2 frisusgra 28319 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
3 usgrav 21363 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
42, 3syl 16 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
54ad2antrr 707 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
65ad2antrr 707 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( V  e.  _V  /\  E  e. 
_V ) )
7 simpr 448 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  a  e.  V )  ->  a  e.  V )
87ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  a  e.  V )
9 simpr 448 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  m  e.  V )
10 eldifsn 3919 . . . . . . . . . . 11  |-  ( b  e.  ( V  \  { a } )  <-> 
( b  e.  V  /\  b  =/=  a
) )
11 simpl 444 . . . . . . . . . . 11  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
b  e.  V )
1210, 11sylbi 188 . . . . . . . . . 10  |-  ( b  e.  ( V  \  { a } )  ->  b  e.  V
)
1312ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  b  e.  V )
148, 9, 133jca 1134 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  (
a  e.  V  /\  m  e.  V  /\  b  e.  V )
)
1514adantr 452 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( a  e.  V  /\  m  e.  V  /\  b  e.  V ) )
16 simpll 731 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  a  =/=  m )
17 necom 2679 . . . . . . . . . . . . . . . . 17  |-  ( b  =/=  a  <->  a  =/=  b )
1817biimpi 187 . . . . . . . . . . . . . . . 16  |-  ( b  =/=  a  ->  a  =/=  b )
1918adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  a  =/=  b )
20 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  m  =/=  b )
2116, 19, 203jca 1134 . . . . . . . . . . . . . 14  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  (
a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) )
2221ex 424 . . . . . . . . . . . . 13  |-  ( ( a  =/=  m  /\  m  =/=  b )  -> 
( b  =/=  a  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2322adantl 453 . . . . . . . . . . . 12  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( b  =/=  a  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2423com12 29 . . . . . . . . . . 11  |-  ( b  =/=  a  ->  (
( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) )  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) ) )
2524adantl 453 . . . . . . . . . 10  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
( ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2610, 25sylbi 188 . . . . . . . . 9  |-  ( b  e.  ( V  \  { a } )  ->  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2726ad2antlr 708 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  (
( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) )  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) ) )
2827imp 419 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) )
29 usgraf1o 21374 . . . . . . . . . 10  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
30 fveq2 5720 . . . . . . . . . . . . . . . . 17  |-  ( ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  ( E `  ( `' E `  { m ,  b } ) ) )
31 simpl 444 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( ( m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  {
a } ) ) )  ->  E : dom  E -1-1-onto-> ran  E )
32 simpll 731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { a ,  m }  e.  ran  E )
33 f1ocnvfv2 6007 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { a ,  m }  e.  ran  E )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m } )
3431, 32, 33syl2an 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m } )
35 simplr 732 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { m ,  b }  e.  ran  E
)
36 f1ocnvfv2 6007 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { m ,  b }  e.  ran  E )  ->  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } )
3731, 35, 36syl2an 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } )
3834, 37eqeq12d 2449 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( ( E `  ( `' E `  { a ,  m } ) )  =  ( E `  ( `' E `  { m ,  b } ) )  <->  { a ,  m }  =  { m ,  b } ) )
39 prcom 3874 . . . . . . . . . . . . . . . . . . . . 21  |-  { m ,  b }  =  { b ,  m }
4039eqeq2i 2445 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { m ,  b }  <->  { a ,  m }  =  { b ,  m } )
41 vex 2951 . . . . . . . . . . . . . . . . . . . . 21  |-  a  e. 
_V
42 vex 2951 . . . . . . . . . . . . . . . . . . . . 21  |-  b  e. 
_V
4341, 42preqr1 3964 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { b ,  m }  ->  a  =  b )
4440, 43sylbi 188 . . . . . . . . . . . . . . . . . . 19  |-  ( { a ,  m }  =  { m ,  b }  ->  a  =  b )
45 df-ne 2600 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( a  =/=  b  <->  -.  a  =  b )
4617, 45bitri 241 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  =/=  a  <->  -.  a  =  b )
47 pm2.21 102 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  a  =  b  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4846, 47sylbi 188 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  =/=  a  ->  (
a  =  b  -> 
( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4948adantl 453 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5010, 49sylbi 188 . . . . . . . . . . . . . . . . . . . . 21  |-  ( b  e.  ( V  \  { a } )  ->  ( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5150adantl 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( m  e.  V  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5251ad2antlr 708 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5344, 52syl5 30 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( {
a ,  m }  =  { m ,  b }  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5438, 53sylbid 207 . . . . . . . . . . . . . . . . 17  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( ( E `  ( `' E `  { a ,  m } ) )  =  ( E `  ( `' E `  { m ,  b } ) )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5530, 54syl5com 28 . . . . . . . . . . . . . . . 16  |-  ( ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  (
( ( E : dom  E -1-1-onto-> ran  E  /\  (
( m  e.  V  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) ) )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
56 df-ne 2600 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  <->  -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } ) )
5756biimpri 198 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) )
5857a1d 23 . . . . . . . . . . . . . . . 16  |-  ( -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( ( ( E : dom  E -1-1-onto-> ran  E  /\  ( ( m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  {
a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) ) )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5955, 58pm2.61i 158 . . . . . . . . . . . . . . 15  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) )
6059, 34, 373jca 1134 . . . . . . . . . . . . . 14  |-  ( ( ( E : dom  E -1-1-onto-> ran 
E  /\  ( (
m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) ) )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) )
6160ex 424 . . . . . . . . . . . . 13  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( ( m  e.  V  /\  a  e.  V )  /\  b  e.  ( V  \  {
a } ) ) )  ->  ( (
( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) ) )
6261expcom 425 . . . . . . . . . . . 12  |-  ( ( ( m  e.  V  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( E : dom  E -1-1-onto-> ran 
E  ->  ( (
( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) ) ) )
6362exp31 588 . . . . . . . . . . 11  |-  ( m  e.  V  ->  (
a  e.  V  -> 
( b  e.  ( V  \  { a } )  ->  ( E : dom  E -1-1-onto-> ran  E  ->  ( ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) ) ) ) ) )
6463com14 84 . . . . . . . . . 10  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( a  e.  V  ->  ( b  e.  ( V  \  { a } )  ->  (
m  e.  V  -> 
( ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) ) ) ) ) )
652, 29, 643syl 19 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( a  e.  V  ->  ( b  e.  ( V  \  {
a } )  -> 
( m  e.  V  ->  ( ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) ) ) ) ) )
6665imp 419 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  a  e.  V )  ->  (
b  e.  ( V 
\  { a } )  ->  ( m  e.  V  ->  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) ) ) ) )
6766imp41 577 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  ( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) )
68 2pthon3v 21596 . . . . . . 7  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  (
a  e.  V  /\  m  e.  V  /\  b  e.  V )  /\  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) )  /\  (
( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  /\  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m }  /\  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } ) )  ->  E. f E. p ( f ( a ( V PathOn  E
) b ) p  /\  ( # `  f
)  =  2 ) )
696, 15, 28, 67, 68syl31anc 1187 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  /\  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) ) )  ->  E. f E. p ( f ( a ( V PathOn  E
) b ) p  /\  ( # `  f
)  =  2 ) )
7069ex 424 . . . . 5  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  (
( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) )  ->  E. f E. p
( f ( a ( V PathOn  E ) b ) p  /\  ( # `  f )  =  2 ) ) )
7170rexlimdva 2822 . . . 4  |-  ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  ->  ( E. m  e.  V  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) )  ->  E. f E. p
( f ( a ( V PathOn  E ) b ) p  /\  ( # `  f )  =  2 ) ) )
7271ralimdva 2776 . . 3  |-  ( ( V FriendGrph  E  /\  a  e.  V )  ->  ( A. b  e.  ( V  \  { a } ) E. m  e.  V  ( ( { a ,  m }  e.  ran  E  /\  {
m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  A. b  e.  ( V  \  { a } ) E. f E. p ( f ( a ( V PathOn  E
) b ) p  /\  ( # `  f
)  =  2 ) ) )
7372ralimdva 2776 . 2  |-  ( V FriendGrph  E  ->  ( A. a  e.  V  A. b  e.  ( V  \  {
a } ) E. m  e.  V  ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. f E. p ( f ( a ( V PathOn  E
) b ) p  /\  ( # `  f
)  =  2 ) ) )
741, 73mpd 15 1  |-  ( V FriendGrph  E  ->  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. f E. p ( f ( a ( V PathOn  E
) b ) p  /\  ( # `  f
)  =  2 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309   {csn 3806   {cpr 3807   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   ran crn 4871   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   2c2 10041   #chash 11610   USGrph cusg 21357   PathOn cpthon 21504   FriendGrph cfrgra 28315
This theorem is referenced by:  frconngra  28348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-usgra 21359  df-wlk 21508  df-trail 21509  df-pth 21510  df-wlkon 21514  df-pthon 21516  df-frgra 28316
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