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Theorem 2pthfrgra 25737
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 25736 . 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 25718 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
3 usgrav 25063 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
42, 3syl 17 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
54ad2antrr 730 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
65ad2antrr 730 . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  a  e.  V )  ->  a  e.  V )
87ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  a  e.  V )
9 simpr 462 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  m  e.  V )
10 eldifsn 4125 . . . . . . . . . . 11  |-  ( b  e.  ( V  \  { a } )  <-> 
( b  e.  V  /\  b  =/=  a
) )
11 simpl 458 . . . . . . . . . . 11  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
b  e.  V )
1210, 11sylbi 198 . . . . . . . . . 10  |-  ( b  e.  ( V  \  { a } )  ->  b  e.  V
)
1312ad2antlr 731 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  b  e.  V )
148, 9, 133jca 1185 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  b  e.  ( V  \  { a } ) )  /\  m  e.  V )  ->  (
a  e.  V  /\  m  e.  V  /\  b  e.  V )
)
1514adantr 466 . . . . . . 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 758 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  a  =/=  m )
17 necom 2689 . . . . . . . . . . . . . . . . 17  |-  ( b  =/=  a  <->  a  =/=  b )
1817biimpi 197 . . . . . . . . . . . . . . . 16  |-  ( b  =/=  a  ->  a  =/=  b )
1918adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  a  =/=  b )
20 simplr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  m  =/=  b )
2116, 19, 203jca 1185 . . . . . . . . . . . . . 14  |-  ( ( ( a  =/=  m  /\  m  =/=  b
)  /\  b  =/=  a )  ->  (
a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) )
2221ex 435 . . . . . . . . . . . . 13  |-  ( ( a  =/=  m  /\  m  =/=  b )  -> 
( b  =/=  a  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2322adantl 467 . . . . . . . . . . . 12  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  -> 
( b  =/=  a  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b
) ) )
2423com12 32 . . . . . . . . . . 11  |-  ( b  =/=  a  ->  (
( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  (
a  =/=  m  /\  m  =/=  b ) )  ->  ( a  =/=  m  /\  a  =/=  b  /\  m  =/=  b ) ) )
2524adantl 467 . . . . . . . . . 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 198 . . . . . . . . 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 731 . . . . . . . 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 430 . . . . . . 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 25083 . . . . . . . . . 10  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
30 fveq2 5881 . . . . . . . . . . . . . . . . 17  |-  ( ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  ( E `  ( `' E `  { m ,  b } ) ) )
31 simpl 458 . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { a ,  m }  e.  ran  E )
33 f1ocnvfv2 6191 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { a ,  m }  e.  ran  E )  ->  ( E `  ( `' E `  { a ,  m } ) )  =  { a ,  m } )
3431, 32, 33syl2an 479 . . . . . . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { m ,  b }  e.  ran  E
)
36 f1ocnvfv2 6191 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { m ,  b }  e.  ran  E )  ->  ( E `  ( `' E `  { m ,  b } ) )  =  { m ,  b } )
3731, 35, 36syl2an 479 . . . . . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . . . . . 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 4078 . . . . . . . . . . . . . . . . . . . . 21  |-  { m ,  b }  =  { b ,  m }
4039eqeq2i 2440 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { m ,  b }  <->  { a ,  m }  =  { b ,  m } )
41 vex 3083 . . . . . . . . . . . . . . . . . . . . 21  |-  a  e. 
_V
42 vex 3083 . . . . . . . . . . . . . . . . . . . . 21  |-  b  e. 
_V
4341, 42preqr1 4174 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { b ,  m }  ->  a  =  b )
4440, 43sylbi 198 . . . . . . . . . . . . . . . . . . 19  |-  ( { a ,  m }  =  { m ,  b }  ->  a  =  b )
45 nesym 2692 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  =/=  a  <->  -.  a  =  b )
46 pm2.21 111 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  a  =  b  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4745, 46sylbi 198 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  =/=  a  ->  (
a  =  b  -> 
( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4847adantl 467 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4910, 48sylbi 198 . . . . . . . . . . . . . . . . . . . . 21  |-  ( b  e.  ( V  \  { a } )  ->  ( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5049adantl 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( m  e.  V  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5150ad2antlr 731 . . . . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . . . 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 2616 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  <->  -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } ) )
5655biimpri 209 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) )
5756a1d 26 . . . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . 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 436 . . . . . . . . . . . 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 607 . . . . . . . . . . 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 91 . . . . . . . . . 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 18 . . . . . . . . 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 430 . . . . . . . 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 596 . . . . . . 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 25332 . . . . . . 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 1267 . . . . . 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 435 . . . . 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 2914 . . . 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 2830 . . 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 2830 . 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 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080    \ cdif 3433   {csn 3998   {cpr 4000   class class class wbr 4423   `'ccnv 4852   dom cdm 4853   ran crn 4854   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   2c2 10666   #chash 12521   USGrph cusg 25055   PathOn cpthon 25230   FriendGrph cfrgra 25714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-usgra 25058  df-wlk 25234  df-trail 25235  df-pth 25236  df-wlkon 25240  df-pthon 25242  df-frgra 25715
This theorem is referenced by:  frconngra  25747
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