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Theorem 2pthfrgra 25138
Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) 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 25137 . 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 25119 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
3 usgrav 24465 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
42, 3syl 16 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
54ad2antrr 725 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
65ad2antrr 725 . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  a  e.  V )  ->  a  e.  V )
87ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  a  e.  V )
9 simpr 461 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  m  e.  V )
10 eldifsn 4157 . . . . . . . . . . 11  |-  ( b  e.  ( V  \  { a } )  <-> 
( b  e.  V  /\  b  =/=  a
) )
11 simpl 457 . . . . . . . . . . 11  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
b  e.  V )
1210, 11sylbi 195 . . . . . . . . . 10  |-  ( b  e.  ( V  \  { a } )  ->  b  e.  V
)
1312ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  b  e.  V )
148, 9, 133jca 1176 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  (
a  e.  V  /\  m  e.  V  /\  b  e.  V )
)
1514adantr 465 . . . . . . 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 753 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  a  =/=  m )
17 necom 2726 . . . . . . . . . . . . . . . . 17  |-  ( b  =/=  a  <->  a  =/=  b )
1817biimpi 194 . . . . . . . . . . . . . . . 16  |-  ( b  =/=  a  ->  a  =/=  b )
1918adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  a  =/=  b )
20 simplr 755 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  m  =/=  b )
2116, 19, 203jca 1176 . . . . . . . . . . . . . 14  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  (
a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) )
2221ex 434 . . . . . . . . . . . . 13  |-  ( ( a  =/=  m  /\  m  =/=  b )  -> 
( b  =/=  a  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2322adantl 466 . . . . . . . . . . . 12  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( b  =/=  a  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2423com12 31 . . . . . . . . . . 11  |-  ( b  =/=  a  ->  (
( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) )  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) ) )
2524adantl 466 . . . . . . . . . 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 195 . . . . . . . . 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 726 . . . . . . . 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 429 . . . . . . 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 24485 . . . . . . . . . 10  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
30 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  ( E `  ( `' E `  { m ,  b } ) ) )
31 simpl 457 . . . . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { a ,  m }  e.  ran  E )
33 f1ocnvfv2 6184 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { a ,  m }  e.  ran  E )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m } )
3431, 32, 33syl2an 477 . . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { m ,  b }  e.  ran  E
)
36 f1ocnvfv2 6184 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { m ,  b }  e.  ran  E )  ->  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } )
3731, 35, 36syl2an 477 . . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . . 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 4110 . . . . . . . . . . . . . . . . . . . . 21  |-  { m ,  b }  =  { b ,  m }
4039eqeq2i 2475 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { m ,  b }  <->  { a ,  m }  =  { b ,  m } )
41 vex 3112 . . . . . . . . . . . . . . . . . . . . 21  |-  a  e. 
_V
42 vex 3112 . . . . . . . . . . . . . . . . . . . . 21  |-  b  e. 
_V
4341, 42preqr1 4206 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { b ,  m }  ->  a  =  b )
4440, 43sylbi 195 . . . . . . . . . . . . . . . . . . 19  |-  ( { a ,  m }  =  { m ,  b }  ->  a  =  b )
45 nesym 2729 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  =/=  a  <->  -.  a  =  b )
46 pm2.21 108 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  a  =  b  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4745, 46sylbi 195 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  =/=  a  ->  (
a  =  b  -> 
( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4847adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4910, 48sylbi 195 . . . . . . . . . . . . . . . . . . . . 21  |-  ( b  e.  ( V  \  { a } )  ->  ( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5049adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( m  e.  V  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5150ad2antlr 726 . . . . . . . . . . . . . . . . . . 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 } ) ) )
5244, 51syl5 32 . . . . . . . . . . . . . . . . . 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 } ) ) )
5338, 52sylbid 215 . . . . . . . . . . . . . . . . 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 } ) ) )
5430, 53syl5com 30 . . . . . . . . . . . . . . . 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 } ) ) )
55 df-ne 2654 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  <->  -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } ) )
5655biimpri 206 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) )
5756a1d 25 . . . . . . . . . . . . . . . 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 } ) ) )
5854, 57pm2.61i 164 . . . . . . . . . . . . . . 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 } ) )
5958, 34, 373jca 1176 . . . . . . . . . . . . . 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 } ) )
6059ex 434 . . . . . . . . . . . . 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 } ) ) )
6160expcom 435 . . . . . . . . . . . 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 } ) ) ) )
6261exp31 604 . . . . . . . . . . 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 } ) ) ) ) ) )
6362com14 88 . . . . . . . . . 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 } ) ) ) ) ) )
642, 29, 633syl 20 . . . . . . . . 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 } ) ) ) ) ) )
6564imp 429 . . . . . . . 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 } ) ) ) ) )
6665imp41 593 . . . . . . 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 } ) )
67 2pthon3v 24733 . . . . . . 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 ) )
686, 15, 28, 66, 67syl31anc 1231 . . . . . 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 ) )
6968ex 434 . . . . 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 ) ) )
7069rexlimdva 2949 . . . 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 ) ) )
7170ralimdva 2865 . . 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 ) ) )
7271ralimdva 2865 . 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 ) ) )
731, 72mpd 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    \ cdif 3468   {csn 4032   {cpr 4034   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   2c2 10606   #chash 12408   USGrph cusg 24457   PathOn cpthon 24631   FriendGrph cfrgra 25115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-usgra 24460  df-wlk 24635  df-trail 24636  df-pth 24637  df-wlkon 24641  df-pthon 24643  df-frgra 25116
This theorem is referenced by:  frconngra  25148
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