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Theorem wlkiswwlksur 24383
Description: Lemma 3 for wlkiswwlkbij2 24385. (Contributed by Alexander van der Vekens, 23-Jul-2018.)
Hypotheses
Ref Expression
wlkiswwlkbij.t  |-  T  =  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }
wlkiswwlkbij.w  |-  W  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
wlkiswwlkbij.f  |-  F  =  ( t  e.  T  |->  ( 2nd `  t
) )
Assertion
Ref Expression
wlkiswwlksur  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T -onto-> W )
Distinct variable groups:    E, p, t, w    N, p, t, w    P, p, t, w   
t, T    V, p, t, w    t, W    w, F    w, T    F, p    T, p    W, p
Allowed substitution hints:    F( t)    W( w)

Proof of Theorem wlkiswwlksur
Dummy variables  f  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkiswwlkbij.t . . . 4  |-  T  =  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }
2 wlkiswwlkbij.w . . . 4  |-  W  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
3 wlkiswwlkbij.f . . . 4  |-  F  =  ( t  e.  T  |->  ( 2nd `  t
) )
41, 2, 3wlkiswwlkfun 24381 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  ->  F : T --> W )
543adant1 1009 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T --> W )
62eleq2i 2540 . . . . . 6  |-  ( p  e.  W  <->  p  e.  { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )
7 fveq1 5858 . . . . . . . 8  |-  ( w  =  p  ->  (
w `  0 )  =  ( p ` 
0 ) )
87eqeq1d 2464 . . . . . . 7  |-  ( w  =  p  ->  (
( w `  0
)  =  P  <->  ( p `  0 )  =  P ) )
98elrab 3256 . . . . . 6  |-  ( p  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  <->  ( p  e.  ( ( V WWalksN  E
) `  N )  /\  ( p `  0
)  =  P ) )
106, 9bitri 249 . . . . 5  |-  ( p  e.  W  <->  ( p  e.  ( ( V WWalksN  E
) `  N )  /\  ( p `  0
)  =  P ) )
11 wlklniswwlkn 24365 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( E. f
( f ( V Walks 
E ) p  /\  ( # `  f )  =  N )  <->  p  e.  ( ( V WWalksN  E
) `  N )
) )
12 df-br 4443 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) p  <->  <. f ,  p >.  e.  ( V Walks  E
) )
13 vex 3111 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
14 vex 3111 . . . . . . . . . . . . . . . . 17  |-  p  e. 
_V
1513, 14op1st 6784 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. f ,  p >. )  =  f
1615eqcomi 2475 . . . . . . . . . . . . . . 15  |-  f  =  ( 1st `  <. f ,  p >. )
1716fveq2i 5862 . . . . . . . . . . . . . 14  |-  ( # `  f )  =  (
# `  ( 1st ` 
<. f ,  p >. ) )
1817eqeq1i 2469 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  N  <->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
1913, 14op2nd 6785 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  <. f ,  p >. )  =  p
2019eqcomi 2475 . . . . . . . . . . . . . . . 16  |-  p  =  ( 2nd `  <. f ,  p >. )
2120fveq1i 5860 . . . . . . . . . . . . . . 15  |-  ( p `
 0 )  =  ( ( 2nd `  <. f ,  p >. ) `  0 )
2221eqeq1i 2469 . . . . . . . . . . . . . 14  |-  ( ( p `  0 )  =  P  <->  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P )
23 opex 4706 . . . . . . . . . . . . . . . 16  |-  <. f ,  p >.  e.  _V
2423a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  <. f ,  p >.  e.  _V )
25 simpll 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  <. f ,  p >.  e.  ( V Walks  E
) )
26 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
2726anim1i 568 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  ( ( # `  ( 1st `  <. f ,  p >. )
)  =  N  /\  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) )
2820a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  p  =  ( 2nd `  <. f ,  p >. ) )
2925, 27, 28jca31 534 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  ( ( <.
f ,  p >.  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  <. f ,  p >. )
)  =  N  /\  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) )  /\  p  =  ( 2nd ` 
<. f ,  p >. ) ) )
30 eleq1 2534 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( u  e.  ( V Walks  E )  <->  <. f ,  p >.  e.  ( V Walks  E ) ) )
31 fveq2 5859 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. f ,  p >.  ->  ( 1st `  u
)  =  ( 1st `  <. f ,  p >. ) )
3231fveq2d 5863 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( # `  ( 1st `  u ) )  =  ( # `  ( 1st `  <. f ,  p >. ) ) )
3332eqeq1d 2464 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( ( # `  ( 1st `  u
) )  =  N  <-> 
( # `  ( 1st `  <. f ,  p >. ) )  =  N ) )
34 fveq2 5859 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. f ,  p >.  ->  ( 2nd `  u
)  =  ( 2nd `  <. f ,  p >. ) )
3534fveq1d 5861 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( ( 2nd `  u ) `  0
)  =  ( ( 2nd `  <. f ,  p >. ) `  0
) )
3635eqeq1d 2464 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( ( ( 2nd `  u ) `
 0 )  =  P  <->  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) )
3733, 36anbi12d 710 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( ( (
# `  ( 1st `  u ) )  =  N  /\  ( ( 2nd `  u ) `
 0 )  =  P )  <->  ( ( # `
 ( 1st `  <. f ,  p >. )
)  =  N  /\  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) ) )
3830, 37anbi12d 710 . . . . . . . . . . . . . . . . . 18  |-  ( u  =  <. f ,  p >.  ->  ( ( u  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  <->  ( <. f ,  p >.  e.  ( V Walks  E )  /\  (
( # `  ( 1st `  <. f ,  p >. ) )  =  N  /\  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) ) ) )
3934eqeq2d 2476 . . . . . . . . . . . . . . . . . 18  |-  ( u  =  <. f ,  p >.  ->  ( p  =  ( 2nd `  u
)  <->  p  =  ( 2nd `  <. f ,  p >. ) ) )
4038, 39anbi12d 710 . . . . . . . . . . . . . . . . 17  |-  ( u  =  <. f ,  p >.  ->  ( ( ( u  e.  ( V Walks 
E )  /\  (
( # `  ( 1st `  u ) )  =  N  /\  ( ( 2nd `  u ) `
 0 )  =  P ) )  /\  p  =  ( 2nd `  u ) )  <->  ( ( <. f ,  p >.  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  <. f ,  p >. )
)  =  N  /\  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) )  /\  p  =  ( 2nd ` 
<. f ,  p >. ) ) ) )
4129, 40syl5ibrcom 222 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  ( u  = 
<. f ,  p >.  -> 
( ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) ) )
4241impancom 440 . . . . . . . . . . . . . . 15  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  u  = 
<. f ,  p >. )  ->  ( ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P  -> 
( ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) ) )
4324, 42spcimedv 3192 . . . . . . . . . . . . . 14  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  ( (
( 2nd `  <. f ,  p >. ) `  0 )  =  P  ->  E. u
( ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) ) )
4422, 43syl5bi 217 . . . . . . . . . . . . 13  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  ( (
p `  0 )  =  P  ->  E. u
( ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) ) )
4512, 18, 44syl2anb 479 . . . . . . . . . . . 12  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  N )  ->  (
( p `  0
)  =  P  ->  E. u ( ( u  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) ) )
4645exlimiv 1693 . . . . . . . . . . 11  |-  ( E. f ( f ( V Walks  E ) p  /\  ( # `  f
)  =  N )  ->  ( ( p `
 0 )  =  P  ->  E. u
( ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) ) )
4711, 46syl6bir 229 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( p  e.  ( ( V WWalksN  E
) `  N )  ->  ( ( p ` 
0 )  =  P  ->  E. u ( ( u  e.  ( V Walks 
E )  /\  (
( # `  ( 1st `  u ) )  =  N  /\  ( ( 2nd `  u ) `
 0 )  =  P ) )  /\  p  =  ( 2nd `  u ) ) ) ) )
4847imp32 433 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u ( ( u  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) )
49 fveq2 5859 . . . . . . . . . . . . . . 15  |-  ( p  =  u  ->  ( 1st `  p )  =  ( 1st `  u
) )
5049fveq2d 5863 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  u ) ) )
5150eqeq1d 2464 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  u ) )  =  N ) )
52 fveq2 5859 . . . . . . . . . . . . . . 15  |-  ( p  =  u  ->  ( 2nd `  p )  =  ( 2nd `  u
) )
5352fveq1d 5861 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  u ) `  0
) )
5453eqeq1d 2464 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  (
( ( 2nd `  p
) `  0 )  =  P  <->  ( ( 2nd `  u ) `  0
)  =  P ) )
5551, 54anbi12d 710 . . . . . . . . . . . 12  |-  ( p  =  u  ->  (
( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P )  <->  ( ( # `
 ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) ) )
5655elrab 3256 . . . . . . . . . . 11  |-  ( u  e.  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) }  <->  ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) ) )
5756anbi1i 695 . . . . . . . . . 10  |-  ( ( u  e.  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) }  /\  p  =  ( 2nd `  u
) )  <->  ( (
u  e.  ( V Walks 
E )  /\  (
( # `  ( 1st `  u ) )  =  N  /\  ( ( 2nd `  u ) `
 0 )  =  P ) )  /\  p  =  ( 2nd `  u ) ) )
5857exbii 1639 . . . . . . . . 9  |-  ( E. u ( u  e. 
{ p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }  /\  p  =  ( 2nd `  u ) )  <->  E. u
( ( u  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) )  /\  p  =  ( 2nd `  u
) ) )
5948, 58sylibr 212 . . . . . . . 8  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u ( u  e. 
{ p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }  /\  p  =  ( 2nd `  u ) ) )
60 df-rex 2815 . . . . . . . 8  |-  ( E. u  e.  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) } p  =  ( 2nd `  u )  <->  E. u ( u  e. 
{ p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }  /\  p  =  ( 2nd `  u ) ) )
6159, 60sylibr 212 . . . . . . 7  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u  e.  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) } p  =  ( 2nd `  u ) )
621rexeqi 3058 . . . . . . 7  |-  ( E. u  e.  T  p  =  ( 2nd `  u
)  <->  E. u  e.  {
p  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) } p  =  ( 2nd `  u
) )
6361, 62sylibr 212 . . . . . 6  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u  e.  T  p  =  ( 2nd `  u ) )
64 fvex 5869 . . . . . . . . 9  |-  ( 2nd `  u )  e.  _V
65 fveq2 5859 . . . . . . . . . 10  |-  ( t  =  u  ->  ( 2nd `  t )  =  ( 2nd `  u
) )
6665, 3fvmptg 5941 . . . . . . . . 9  |-  ( ( u  e.  T  /\  ( 2nd `  u )  e.  _V )  -> 
( F `  u
)  =  ( 2nd `  u ) )
6764, 66mpan2 671 . . . . . . . 8  |-  ( u  e.  T  ->  ( F `  u )  =  ( 2nd `  u
) )
6867eqeq2d 2476 . . . . . . 7  |-  ( u  e.  T  ->  (
p  =  ( F `
 u )  <->  p  =  ( 2nd `  u ) ) )
6968rexbiia 2959 . . . . . 6  |-  ( E. u  e.  T  p  =  ( F `  u )  <->  E. u  e.  T  p  =  ( 2nd `  u ) )
7063, 69sylibr 212 . . . . 5  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u  e.  T  p  =  ( F `  u ) )
7110, 70sylan2b 475 . . . 4  |-  ( ( V USGrph  E  /\  p  e.  W )  ->  E. u  e.  T  p  =  ( F `  u ) )
7271ralrimiva 2873 . . 3  |-  ( V USGrph  E  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
73723ad2ant1 1012 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
74 dffo3 6029 . 2  |-  ( F : T -onto-> W  <->  ( F : T --> W  /\  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) ) )
755, 73, 74sylanbrc 664 1  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T -onto-> W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   A.wral 2809   E.wrex 2810   {crab 2813   _Vcvv 3108   <.cop 4028   class class class wbr 4442    |-> cmpt 4500   -->wf 5577   -onto->wfo 5579   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775   0cc0 9483   NN0cn0 10786   #chash 12362   USGrph cusg 23995   Walks cwalk 24162   WWalksN cwwlkn 24342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-usgra 23998  df-wlk 24172  df-wwlk 24343  df-wwlkn 24344
This theorem is referenced by:  wlkiswwlkbij  24384
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