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Theorem frg2woteq 25786
Description: There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 14-Feb-2018.)
Assertion
Ref Expression
frg2woteq  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) )

Proof of Theorem frg2woteq
Dummy variables  c 
d  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2wlkonot3v 25601 . . . 4  |-  ( P  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) ) )
21adantr 466 . . 3  |-  ( ( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) ) )
3 el2wlkonot 25595 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( P  e.  ( A
( V 2WalksOnOt  E ) B )  <->  E. c  e.  V  ( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) ) ) )
4 pm3.22 450 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  A  e.  V
) )
54anim2i 571 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V
) ) )
6 el2wlkonot 25595 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V
) )  ->  ( Q  e.  ( B
( V 2WalksOnOt  E ) A )  <->  E. d  e.  V  ( Q  =  <. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) ) ) )
75, 6syl 17 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( Q  e.  ( B
( V 2WalksOnOt  E ) A )  <->  E. d  e.  V  ( Q  =  <. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) ) ) )
83, 7anbi12d 715 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  (
( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  <->  ( E. c  e.  V  ( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  c  =  ( p `  1
)  /\  B  =  ( p `  2
) ) ) )  /\  E. d  e.  V  ( Q  = 
<. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) ) ) ) )
983adant3 1025 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  <->  ( E. c  e.  V  ( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  c  =  ( p `  1
)  /\  B  =  ( p `  2
) ) ) )  /\  E. d  e.  V  ( Q  = 
<. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) ) ) ) )
1053adant3 1025 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V
) ) )
1110adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) )  /\  c  e.  V )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V
) ) )
1211ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V )
) )
13 el2wlkonotot 25599 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V
) )  ->  ( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) ) )
1413bicomd 204 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( B  e.  V  /\  A  e.  V
) )  ->  ( E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( B  =  ( p ` 
0 )  /\  d  =  ( p ` 
1 )  /\  A  =  ( p ` 
2 ) ) )  <->  <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) ) )
1512, 14syl 17 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) )  <->  <. B , 
d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) ) )
16 3simpa 1002 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) ) )
1716ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) )  /\  c  e.  V )  /\  d  e.  V )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) ) )
1817ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
) )
19 el2wlkonotot 25599 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) ) )
2019bicomd 204 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  c  =  ( p ` 
1 )  /\  B  =  ( p ` 
2 ) ) )  <->  <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) ) )
2118, 20syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  c  =  ( p `  1
)  /\  B  =  ( p `  2
) ) )  <->  <. A , 
c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) ) )
22 frg2woteqm 25785 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  d  =  c ) )
23 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  P )  =  ( 1st `  <. A ,  c ,  B >. ) )
2423fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2524adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2625adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
27 vex 3083 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  c  e. 
_V
28 ot1stg 6821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  c  e.  _V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
2927, 28mp3an2 1348 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
)  =  A )
3029ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
3130adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
32 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 2nd `  Q )  =  ( 2nd `  <. B ,  d ,  A >. ) )
3332ad2antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd `  Q )  =  ( 2nd `  <. B , 
d ,  A >. ) )
3433adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  Q )  =  ( 2nd `  <. B ,  d ,  A >. ) )
35 ot3rdg 6823 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( A  e.  V  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3635adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3736ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. B ,  d ,  A >. )  =  A )
3837adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3934, 38eqtr2d 2464 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  A  =  ( 2nd `  Q
) )
4026, 31, 393eqtrd 2467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q ) )
41 eqidd 2423 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) ) )
42 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  Q )  =  ( 1st `  <. B ,  d ,  A >. ) )
4342fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4443ad2antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4544adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
46 simpr 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  B  e.  V )
47 vex 3083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  d  e. 
_V
4847a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  d  e.  _V )
49 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
5046, 48, 493jca 1185 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  d  e.  _V  /\  A  e.  V ) )
5150ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( B  e.  V  /\  d  e.  _V  /\  A  e.  V ) )
5251adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( B  e.  V  /\  d  e.  _V  /\  A  e.  V ) )
53 ot1stg 6821 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( B  e.  V  /\  d  e.  _V  /\  A  e.  V )  ->  ( 1st `  ( 1st `  <. B ,  d ,  A >. ) )  =  B )
5452, 53syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  <. B ,  d ,  A >. ) )  =  B )
55 ot3rdg 6823 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( B  e.  V  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5655adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5756ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. A ,  c ,  B >. )  =  B )
5857adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
59 simpr 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  P  =  <. A ,  c ,  B >. )
6059adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  P  =  <. A ,  c ,  B >. )
6160eqcomd 2430 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  <. A , 
c ,  B >.  =  P )
6261fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  ( 2nd `  P
) )
6358, 62eqtr3d 2465 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  P
) )
6445, 54, 633eqtrd 2467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 2nd `  P ) )
6540, 41, 643jca 1185 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) )
6665ex 435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( d  =  c  ->  (
( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) )
6722, 66syl6 34 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( (
( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) )
6867expd 437 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
6968com14 91 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B , 
d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  ( ( V FriendGrph  E  /\  A  =/=  B
)  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
7069ex 435 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  Q  =  <. B ,  d ,  A >. )  ->  ( P  =  <. A , 
c ,  B >.  -> 
( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( V FriendGrph  E  /\  A  =/=  B )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) ) )
7170ex 435 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( Q  =  <. B ,  d ,  A >.  ->  ( P  = 
<. A ,  c ,  B >.  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( V FriendGrph  E  /\  A  =/=  B )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) ) ) )
72713ad2ant2 1027 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  ( Q  =  <. B , 
d ,  A >.  -> 
( P  =  <. A ,  c ,  B >.  ->  ( <. A , 
c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B , 
d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  ( ( V FriendGrph  E  /\  A  =/=  B
)  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) ) ) )
7372ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) )  /\  c  e.  V )  /\  d  e.  V )  ->  ( Q  =  <. B , 
d ,  A >.  -> 
( P  =  <. A ,  c ,  B >.  ->  ( <. A , 
c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B , 
d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  ( ( V FriendGrph  E  /\  A  =/=  B
)  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) ) ) )
7473imp31 433 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  ->  ( <. B , 
d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  ( ( V FriendGrph  E  /\  A  =/=  B
)  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
7521, 74sylbid 218 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  c  =  ( p `  1
)  /\  B  =  ( p `  2
) ) )  -> 
( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( V FriendGrph  E  /\  A  =/=  B )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
7675expimpd 606 . . . . . . . . . . 11  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  ->  ( ( P  = 
<. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( V FriendGrph  E  /\  A  =/=  B )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
7776com23 81 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  ->  ( <. B ,  d ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A )  ->  (
( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
7815, 77sylbid 218 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  /\  c  e.  V )  /\  d  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) )  ->  (
( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
7978expimpd 606 . . . . . . . 8  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) )  /\  c  e.  V )  /\  d  e.  V )  ->  (
( Q  =  <. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) )  -> 
( ( P  = 
<. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
8079rexlimdva 2914 . . . . . . 7  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) )  /\  c  e.  V )  ->  ( E. d  e.  V  ( Q  =  <. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) )  -> 
( ( P  = 
<. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
8180com23 81 . . . . . 6  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )  /\  P  e.  (
( V  X.  V
)  X.  V ) )  /\  c  e.  V )  ->  (
( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( E. d  e.  V  ( Q  = 
<. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
8281rexlimdva 2914 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  ( E. c  e.  V  ( P  =  <. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  -> 
( E. d  e.  V  ( Q  = 
<. B ,  d ,  A >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) ) )
8382impd 432 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( E. c  e.  V  ( P  = 
<. A ,  c ,  B >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  c  =  ( p `  1 )  /\  B  =  ( p `  2 ) ) ) )  /\  E. d  e.  V  ( Q  =  <. B , 
d ,  A >.  /\ 
E. f E. p
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( B  =  ( p `  0 )  /\  d  =  ( p `  1 )  /\  A  =  ( p `  2 ) ) ) ) )  -> 
( ( V FriendGrph  E  /\  A  =/=  B )  -> 
( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) )
849, 83sylbid 218 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
)  /\  P  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) ) )
852, 84mpcom 37 . 2  |-  ( ( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q
)  /\  ( 2nd `  ( 1st `  P
) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) )
8685com12 32 1  |-  ( ( V FriendGrph  E  /\  A  =/= 
B )  ->  (
( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) )  /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   E.wrex 2772   _Vcvv 3080   <.cotp 4006   class class class wbr 4423    X. cxp 4851   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   0cc0 9546   1c1 9547   2c2 10666   #chash 12521   Walks cwalk 25224   2WalksOnOt c2wlkonot 25581   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-ot 4007  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-wlkon 25240  df-2wlkonot 25584  df-frgra 25715
This theorem is referenced by: (None)
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