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Theorem 2pthfrgra 24834
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 24833 . 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 24815 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
3 usgrav 24161 . . . . . . . . . 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 4158 . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . 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 24181 . . . . . . . . . 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 6182 . . . . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( { a ,  m }  e.  ran  E  /\  { m ,  b }  e.  ran  E )  /\  ( a  =/=  m  /\  m  =/=  b ) )  ->  { m ,  b }  e.  ran  E
)
36 f1ocnvfv2 6182 . . . . . . . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . . . . . . 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 4111 . . . . . . . . . . . . . . . . . . . . 21  |-  { m ,  b }  =  { b ,  m }
4039eqeq2i 2485 . . . . . . . . . . . . . . . . . . . 20  |-  ( { a ,  m }  =  { m ,  b }  <->  { a ,  m }  =  { b ,  m } )
41 vex 3121 . . . . . . . . . . . . . . . . . . . . 21  |-  a  e. 
_V
42 vex 3121 . . . . . . . . . . . . . . . . . . . . 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 df-ne 2664 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( a  =/=  b  <->  -.  a  =  b )
4617, 45bitri 249 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  =/=  a  <->  -.  a  =  b )
47 pm2.21 108 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  a  =  b  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4846, 47sylbi 195 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  =/=  a  ->  (
a  =  b  -> 
( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
4948adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( b  e.  V  /\  b  =/=  a )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5010, 49sylbi 195 . . . . . . . . . . . . . . . . . . . . 21  |-  ( b  e.  ( V  \  { a } )  ->  ( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5150adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( m  e.  V  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( a  =  b  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) ) )
5251ad2antlr 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 } ) ) )
5344, 52syl5 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 } ) ) )
5438, 53sylbid 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 } ) ) )
5530, 54syl5com 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 } ) ) )
56 df-ne 2664 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } )  <->  -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } ) )
5756biimpri 206 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( `' E `  { a ,  m } )  =  ( `' E `  { m ,  b } )  ->  ( `' E `  { a ,  m } )  =/=  ( `' E `  { m ,  b } ) )
5857a1d 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 } ) ) )
5955, 58pm2.61i 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 } ) )
6059, 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 } ) )
6160ex 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 } ) ) )
6261expcom 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 } ) ) ) )
6362exp31 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 } ) ) ) ) ) )
6463com14 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 } ) ) ) ) ) )
652, 29, 643syl 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 } ) ) ) ) ) )
6665imp 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 } ) ) ) ) )
6766imp41 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 } ) )
68 2pthon3v 24429 . . . . . . 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 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 ) )
7069ex 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 ) ) )
7170rexlimdva 2959 . . . 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 2875 . . 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 2875 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118    \ cdif 3478   {csn 4033   {cpr 4035   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   ran crn 5006   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   2c2 10597   #chash 12385   USGrph cusg 24153   PathOn cpthon 24327   FriendGrph cfrgra 24811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-usgra 24156  df-wlk 24331  df-trail 24332  df-pth 24333  df-wlkon 24337  df-pthon 24339  df-frgra 24812
This theorem is referenced by:  frconngra  24844
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