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Theorem frg2woteq 25464
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 25279 . . . 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 463 . . 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 25273 . . . . . 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 pm3.22 447 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  A  e.  V
) )
54anim2i 567 . . . . . . 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 25273 . . . . . . 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 709 . . . . 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 1017 . . . 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 1017 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 724 . . . . . . . . . . 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 25277 . . . . . . . . . . . 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 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 994 . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 25277 . . . . . . . . . . . . . . 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 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 25463 . . . . . . . . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  P )  =  ( 1st `  <. A ,  c ,  B >. ) )
2423fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( P  =  <. A ,  c ,  B >.  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2524adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  P
) )  =  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
) )
2625adantl 464 . . . . . . . . . . . . . . . . . . . . . . . 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 3061 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  c  e. 
_V
28 ot1stg 6797 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  c  e.  _V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
2927, 28mp3an2 1314 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. )
)  =  A )
3029ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  <. A ,  c ,  B >. ) )  =  A )
3130adantl 464 . . . . . . . . . . . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 2nd `  Q )  =  ( 2nd `  <. B ,  d ,  A >. ) )
3332ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd `  Q )  =  ( 2nd `  <. B , 
d ,  A >. ) )
3433adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . 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 6799 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( A  e.  V  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3635adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3736ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. B ,  d ,  A >. )  =  A )
3837adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. B ,  d ,  A >. )  =  A )
3934, 38eqtr2d 2444 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  A  =  ( 2nd `  Q
) )
4026, 31, 393eqtrd 2447 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 1st `  ( 1st `  P
) )  =  ( 2nd `  Q ) )
41 eqidd 2403 . . . . . . . . . . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  Q )  =  ( 1st `  <. B ,  d ,  A >. ) )
4342fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Q  =  <. B ,  d ,  A >.  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4443ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 1st `  ( 1st `  Q
) )  =  ( 1st `  ( 1st `  <. B ,  d ,  A >. )
) )
4544adantl 464 . . . . . . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  B  e.  V )
47 vex 3061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  d  e. 
_V
4847a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  d  e.  _V )
49 simpl 455 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
5046, 48, 493jca 1177 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( B  e.  V  /\  d  e.  _V  /\  A  e.  V ) )
5150ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . . . . . . . . . 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 6797 . . . . . . . . . . . . . . . . . . . . . . . . 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 6799 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( B  e.  V  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5655adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
5756ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  ( 2nd ` 
<. A ,  c ,  B >. )  =  B )
5857adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  ( 2nd `  <. A ,  c ,  B >. )  =  B )
59 simpr 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. )  ->  P  =  <. A ,  c ,  B >. )
6059adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  P  =  <. A ,  c ,  B >. )
6160eqcomd 2410 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  <. A , 
c ,  B >.  =  P )
6261fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( d  =  c  /\  ( ( ( A  e.  V  /\  B  e.  V )  /\  Q  =  <. B ,  d ,  A >. )  /\  P  =  <. A ,  c ,  B >. ) )  ->  B  =  ( 2nd `  P
) )
6445, 54, 633eqtrd 2447 . . . . . . . . . . . . . . . . . . . . . . 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 1177 . . . . . . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . . . . . . 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 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
) ) ) ) )
6867expd 434 . . . . . . . . . . . . . . . . . . 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 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
) ) ) ) ) )
7069ex 432 . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . 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 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
) ) ) ) ) )
7675expimpd 601 . . . . . . . . . . 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 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
) ) ) ) ) )
7815, 77sylbid 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
) ) ) ) ) )
7978expimpd 601 . . . . . . . 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 2895 . . . . . . 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 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
) ) ) ) ) )
8281rexlimdva 2895 . . . . 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 429 . . . 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 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
) ) ) ) )
852, 84mpcom 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
) ) ) )
8685com12 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   E.wrex 2754   _Vcvv 3058   <.cotp 3979   class class class wbr 4394    X. cxp 4820   ` cfv 5568  (class class class)co 6277   1stc1st 6781   2ndc2nd 6782   0cc0 9521   1c1 9522   2c2 10625   #chash 12450   Walks cwalk 24902   2WalksOnOt c2wlkonot 25259   FriendGrph cfrgra 25392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-usgra 24737  df-wlk 24912  df-wlkon 24918  df-2wlkonot 25262  df-frgra 25393
This theorem is referenced by: (None)
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