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Theorem frg2woteq 30824
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 30565 . . . 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 465 . . 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 30559 . . . . . 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 pm3.22 449 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  A  e.  V
) )
54anim2i 569 . . . . . . 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 30559 . . . . . . 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 710 . . . . 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 1008 . . . 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 1008 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 725 . . . . . . . . . . 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 30563 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 30563 . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . 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 30823 . . . . . . . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  P )  =  ( 1st `  <. A ,  c ,  B >. ) )
2423fveq2d 5806 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2524adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2625adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 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 3081 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  c  e. 
_V
28 ot1stg 6704 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  c  e.  _V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
2927, 28mp3an2 1303 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
)  =  A )
3029ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
3130adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 2nd `  Q )  =  ( 2nd `  <. B ,  d ,  A >. ) )
3332ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd `  Q )  =  ( 2nd `  <. B , 
d ,  A >. ) )
3433adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . 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 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( A  e.  V  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3635adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3736ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. B ,  d ,  A >. )  =  A )
3837adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3934, 38eqtr2d 2496 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  A  =  ( 2nd `  Q
) )
4026, 31, 393eqtrd 2499 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q ) )
41 eqidd 2455 . . . . . . . . . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  Q )  =  ( 1st `  <. B ,  d ,  A >. ) )
4342fveq2d 5806 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4443ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4544adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  B  e.  V )
47 vex 3081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  d  e. 
_V
4847a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  d  e.  _V )
49 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
5046, 48, 493jca 1168 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  d  e.  _V  /\  A  e.  V ) )
5150ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . . . 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 6704 . . . . . . . . . . . . . . . . . . . . . . . . 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 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( B  e.  V  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5655adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5756ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. A ,  c ,  B >. )  =  B )
5857adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5958eqcomd 2462 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  <. A ,  c ,  B >. ) )
60 simpr 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  P  =  <. A ,  c ,  B >. )
6160adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  P  =  <. A ,  c ,  B >. )
6261eqcomd 2462 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  <. A , 
c ,  B >.  =  P )
6362fveq2d 5806 . . . . . . . . . . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  P
) )
6545, 54, 643eqtrd 2499 . . . . . . . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . . . . . . 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
) ) ) ) )
6968expd 436 . . . . . . . . . . . . . . . . . . 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 88 . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 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 603 . . . . . . . . . . 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 78 . . . . . . . . . 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 215 . . . . . . . . 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 603 . . . . . . . 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 2947 . . . . . . 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 78 . . . . . 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 2947 . . . . 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
) ) ) ) ) )
8483impd 431 . . . 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 215 . . 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 36 . 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 31 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 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   E.wrex 2800   _Vcvv 3078   <.cotp 3996   class class class wbr 4403    X. cxp 4949   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   0cc0 9397   1c1 9398   2c2 10486   #chash 12224   Walks cwalk 23584   2WalksOnOt c2wlkonot 30545   FriendGrph cfrgra 30751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-ot 3997  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-usgra 23445  df-wlk 23594  df-wlkon 23600  df-2wlkonot 30548  df-frgra 30752
This theorem is referenced by: (None)
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