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Theorem frg2woteq 28163
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 28072 . . . 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 452 . . 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 28066 . . . . . 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 pm3.22 437 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  A  e.  V
) )
54anim2i 553 . . . . . . 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 28066 . . . . . . 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 16 . . . . . 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 692 . . . . 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 977 . . . 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 977 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 707 . . . . . . . . . . 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 28070 . . . . . . . . . . . 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 193 . . . . . . . . . . 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 16 . . . . . . . . . 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 954 . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . 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 28070 . . . . . . . . . . . . . . 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 193 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 28162 . . . . . . . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  P )  =  ( 1st `  <. A ,  c ,  B >. ) )
2423fveq2d 5691 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2524adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2625adantl 453 . . . . . . . . . . . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  c  e. 
_V
28 ot1stg 6320 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  c  e.  _V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
2927, 28mp3an2 1267 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
)  =  A )
3029ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
3130adantl 453 . . . . . . . . . . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 2nd `  Q )  =  ( 2nd `  <. B ,  d ,  A >. ) )
3332ad2antlr 708 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd `  Q )  =  ( 2nd `  <. B , 
d ,  A >. ) )
3433adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( A  e.  V  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3635adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3736ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. B ,  d ,  A >. )  =  A )
3837adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3934, 38eqtr2d 2437 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  A  =  ( 2nd `  Q
) )
4026, 31, 393eqtrd 2440 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q ) )
41 eqidd 2405 . . . . . . . . . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  Q )  =  ( 1st `  <. B ,  d ,  A >. ) )
4342fveq2d 5691 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4443ad2antlr 708 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4544adantl 453 . . . . . . . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  B  e.  V )
47 vex 2919 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  d  e. 
_V
4847a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  d  e.  _V )
49 simpl 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
5046, 48, 493jca 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  d  e.  _V  /\  A  e.  V ) )
5150ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( B  e.  V  /\  d  e.  _V  /\  A  e.  V )  ->  ( 1st `  ( 1st `  <. B ,  d ,  A >. ) )  =  B )
5452, 53syl 16 . . . . . . . . . . . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( B  e.  V  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5655adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5756ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. A ,  c ,  B >. )  =  B )
5857adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5958eqcomd 2409 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  <. A ,  c ,  B >. ) )
60 simpr 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  P  =  <. A ,  c ,  B >. )
6160adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  P  =  <. A ,  c ,  B >. )
6261eqcomd 2409 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  <. A , 
c ,  B >.  =  P )
6362fveq2d 5691 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  ( 2nd `  P
) )
6459, 63eqtrd 2436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  P
) )
6545, 54, 643eqtrd 2440 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 2nd `  P ) )
6640, 41, 653jca 1134 . . . . . . . . . . . . . . . . . . . . . 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
) ) )
6766ex 424 . . . . . . . . . . . . . . . . . . . . 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
) ) ) )
6822, 67syl6 31 . . . . . . . . . . . . . . . . . . . 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
) ) ) ) )
6968exp3a 426 . . . . . . . . . . . . . . . . . . 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
) ) ) ) ) )
7069com14 84 . . . . . . . . . . . . . . . . . 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
) ) ) ) ) )
7170ex 424 . . . . . . . . . . . . . . . . 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
) ) ) ) ) ) )
7271ex 424 . . . . . . . . . . . . . . . 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
) ) ) ) ) ) ) )
73723ad2ant2 979 . . . . . . . . . . . . . . 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
) ) ) ) ) ) ) )
7473ad2antrr 707 . . . . . . . . . . . . . 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
) ) ) ) ) ) ) )
7574imp31 422 . . . . . . . . . . . . 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
) ) ) ) ) )
7621, 75sylbid 207 . . . . . . . . . . . 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
) ) ) ) ) )
7776expimpd 587 . . . . . . . . . . 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
) ) ) ) ) )
7877com23 74 . . . . . . . . . 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
) ) ) ) ) )
7915, 78sylbid 207 . . . . . . . . 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
) ) ) ) ) )
8079expimpd 587 . . . . . . . 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
) ) ) ) ) )
8180rexlimdva 2790 . . . . . . 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
) ) ) ) ) )
8281com23 74 . . . . . 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
) ) ) ) ) )
8382rexlimdva 2790 . . . . 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
) ) ) ) ) )
8483imp3a 421 . . . 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
) ) ) ) )
859, 84sylbid 207 . . 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
) ) ) ) )
862, 85mpcom 34 . 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
) ) ) )
8786com12 29 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   _Vcvv 2916   <.cotp 3778   class class class wbr 4172    X. cxp 4835   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   0cc0 8946   1c1 8947   2c2 10005   #chash 11573   Walks cwalk 21459   2WalksOnOt c2wlkonot 28052   FriendGrph cfrgra 28092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-usgra 21320  df-wlk 21469  df-wlkon 21475  df-2wlkonot 28055  df-frgra 28093
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