Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlkiswwlksur Structured version   Unicode version

Theorem wlkiswwlksur 30349
Description: Lemma 3 for wlkiswwlkbij2 30351. (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 30347 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  ->  F : T --> W )
543adant1 1006 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T --> W )
62eleq2i 2506 . . . . . 6  |-  ( p  e.  W  <->  p  e.  { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )
7 fveq1 5689 . . . . . . . 8  |-  ( w  =  p  ->  (
w `  0 )  =  ( p ` 
0 ) )
87eqeq1d 2450 . . . . . . 7  |-  ( w  =  p  ->  (
( w `  0
)  =  P  <->  ( p `  0 )  =  P ) )
98elrab 3116 . . . . . 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 30333 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( E. f
( f ( V Walks 
E ) p  /\  ( # `  f )  =  N )  <->  p  e.  ( ( V WWalksN  E
) `  N )
) )
12 df-br 4292 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) p  <->  <. f ,  p >.  e.  ( V Walks  E
) )
13 vex 2974 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
14 vex 2974 . . . . . . . . . . . . . . . . 17  |-  p  e. 
_V
1513, 14op1st 6584 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. f ,  p >. )  =  f
1615eqcomi 2446 . . . . . . . . . . . . . . 15  |-  f  =  ( 1st `  <. f ,  p >. )
1716fveq2i 5693 . . . . . . . . . . . . . 14  |-  ( # `  f )  =  (
# `  ( 1st ` 
<. f ,  p >. ) )
1817eqeq1i 2449 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  N  <->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
1913, 14op2nd 6585 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  <. f ,  p >. )  =  p
2019eqcomi 2446 . . . . . . . . . . . . . . . 16  |-  p  =  ( 2nd `  <. f ,  p >. )
2120fveq1i 5691 . . . . . . . . . . . . . . 15  |-  ( p `
 0 )  =  ( ( 2nd `  <. f ,  p >. ) `  0 )
2221eqeq1i 2449 . . . . . . . . . . . . . 14  |-  ( ( p `  0 )  =  P  <->  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P )
23 opex 4555 . . . . . . . . . . . . . . . 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 2502 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( u  e.  ( V Walks  E )  <->  <. f ,  p >.  e.  ( V Walks  E ) ) )
31 fveq2 5690 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. f ,  p >.  ->  ( 1st `  u
)  =  ( 1st `  <. f ,  p >. ) )
3231fveq2d 5694 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( # `  ( 1st `  u ) )  =  ( # `  ( 1st `  <. f ,  p >. ) ) )
3332eqeq1d 2450 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( ( # `  ( 1st `  u
) )  =  N  <-> 
( # `  ( 1st `  <. f ,  p >. ) )  =  N ) )
34 fveq2 5690 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. f ,  p >.  ->  ( 2nd `  u
)  =  ( 2nd `  <. f ,  p >. ) )
3534fveq1d 5692 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( ( 2nd `  u ) `  0
)  =  ( ( 2nd `  <. f ,  p >. ) `  0
) )
3635eqeq1d 2450 . . . . . . . . . . . . . . . . . . . 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 2453 . . . . . . . . . . . . . . . . . 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 3055 . . . . . . . . . . . . . 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 1688 . . . . . . . . . . 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 5690 . . . . . . . . . . . . . . 15  |-  ( p  =  u  ->  ( 1st `  p )  =  ( 1st `  u
) )
5049fveq2d 5694 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  u ) ) )
5150eqeq1d 2450 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  u ) )  =  N ) )
52 fveq2 5690 . . . . . . . . . . . . . . 15  |-  ( p  =  u  ->  ( 2nd `  p )  =  ( 2nd `  u
) )
5352fveq1d 5692 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  u ) `  0
) )
5453eqeq1d 2450 . . . . . . . . . . . . 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 3116 . . . . . . . . . . 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 1634 . . . . . . . . 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 2720 . . . . . . . 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 2921 . . . . . . 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 5700 . . . . . . . . 9  |-  ( 2nd `  u )  e.  _V
65 fveq2 5690 . . . . . . . . . 10  |-  ( t  =  u  ->  ( 2nd `  t )  =  ( 2nd `  u
) )
6665, 3fvmptg 5771 . . . . . . . . 9  |-  ( ( u  e.  T  /\  ( 2nd `  u )  e.  _V )  -> 
( F `  u
)  =  ( 2nd `  u ) )
6764, 66mpan2 671 . . . . . . . 8  |-  ( u  e.  T  ->  ( F `  u )  =  ( 2nd `  u
) )
6867eqeq2d 2453 . . . . . . 7  |-  ( u  e.  T  ->  (
p  =  ( F `
 u )  <->  p  =  ( 2nd `  u ) ) )
6968rexbiia 2747 . . . . . 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 2798 . . 3  |-  ( V USGrph  E  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
73723ad2ant1 1009 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
74 dffo3 5857 . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 2971   <.cop 3882   class class class wbr 4291    e. cmpt 4349   -->wf 5413   -onto->wfo 5415   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   0cc0 9281   NN0cn0 10578   #chash 12102   USGrph cusg 23263   Walks cwalk 23404   WWalksN cwwlkn 30310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-hash 12103  df-word 12228  df-usgra 23265  df-wlk 23414  df-wwlk 30311  df-wwlkn 30312
This theorem is referenced by:  wlkiswwlkbij  30350
  Copyright terms: Public domain W3C validator