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Theorem frg2woteq 28449
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 28342 . . . 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 28336 . . . . . 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 28336 . . . . . . 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 28340 . . . . . . . . . . . 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 28340 . . . . . . . . . . . . . . 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 28448 . . . . . . . . . . . . . . . . . . . . 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 5728 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  P )  =  ( 1st `  <. A ,  c ,  B >. ) )
2423fveq2d 5732 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2959 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  c  e. 
_V
28 ot1stg 6361 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5728 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  A  =  ( 2nd `  Q
) )
4026, 31, 393eqtrd 2472 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q ) )
41 eqidd 2437 . . . . . . . . . . . . . . . . . . . . . . 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 5728 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  Q )  =  ( 1st `  <. B ,  d ,  A >. ) )
4342fveq2d 5732 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6361 . . . . . . . . . . . . . . . . . . . . . . . . 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 6363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  <. A , 
c ,  B >.  =  P )
6362fveq2d 5732 . . . . . . . . . . . . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  P
) )
6545, 54, 643eqtrd 2472 . . . . . . . . . . . . . . . . . . . . . . 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 2830 . . . . . . 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 2830 . . . . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   _Vcvv 2956   <.cotp 3818   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   0cc0 8990   1c1 8991   2c2 10049   #chash 11618   Walks cwalk 21506   2WalksOnOt c2wlkonot 28322   FriendGrph cfrgra 28378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-usgra 21367  df-wlk 21516  df-wlkon 21522  df-2wlkonot 28325  df-frgra 28379
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