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Theorem wlkiswwlksur 24840
Description: Lemma 3 for wlkiswwlkbij2 24842. (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 24838 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  ->  F : T --> W )
543adant1 1012 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T --> W )
6 fveq1 5773 . . . . . . 7  |-  ( w  =  p  ->  (
w `  0 )  =  ( p ` 
0 ) )
76eqeq1d 2384 . . . . . 6  |-  ( w  =  p  ->  (
( w `  0
)  =  P  <->  ( p `  0 )  =  P ) )
87, 2elrab2 3184 . . . . 5  |-  ( p  e.  W  <->  ( p  e.  ( ( V WWalksN  E
) `  N )  /\  ( p `  0
)  =  P ) )
9 wlklniswwlkn 24822 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( E. f
( f ( V Walks 
E ) p  /\  ( # `  f )  =  N )  <->  p  e.  ( ( V WWalksN  E
) `  N )
) )
10 df-br 4368 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) p  <->  <. f ,  p >.  e.  ( V Walks  E
) )
11 vex 3037 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
12 vex 3037 . . . . . . . . . . . . . . . . 17  |-  p  e. 
_V
1311, 12op1st 6707 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. f ,  p >. )  =  f
1413eqcomi 2395 . . . . . . . . . . . . . . 15  |-  f  =  ( 1st `  <. f ,  p >. )
1514fveq2i 5777 . . . . . . . . . . . . . 14  |-  ( # `  f )  =  (
# `  ( 1st ` 
<. f ,  p >. ) )
1615eqeq1i 2389 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  N  <->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
1711, 12op2nd 6708 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  <. f ,  p >. )  =  p
1817eqcomi 2395 . . . . . . . . . . . . . . . 16  |-  p  =  ( 2nd `  <. f ,  p >. )
1918fveq1i 5775 . . . . . . . . . . . . . . 15  |-  ( p `
 0 )  =  ( ( 2nd `  <. f ,  p >. ) `  0 )
2019eqeq1i 2389 . . . . . . . . . . . . . 14  |-  ( ( p `  0 )  =  P  <->  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P )
21 opex 4626 . . . . . . . . . . . . . . . 16  |-  <. f ,  p >.  e.  _V
2221a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  <. f ,  p >.  e.  _V )
23 simpll 751 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  <. f ,  p >.  e.  ( V Walks  E
) )
24 simpr 459 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
2524anim1i 566 . . . . . . . . . . . . . . . . . 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 ) )
2618a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  /\  ( ( 2nd `  <. f ,  p >. ) `  0
)  =  P )  ->  p  =  ( 2nd `  <. f ,  p >. ) )
2723, 25, 26jca31 532 . . . . . . . . . . . . . . . . 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 >. ) ) )
28 eleq1 2454 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( u  e.  ( V Walks  E )  <->  <. f ,  p >.  e.  ( V Walks  E ) ) )
29 fveq2 5774 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. f ,  p >.  ->  ( 1st `  u
)  =  ( 1st `  <. f ,  p >. ) )
3029fveq2d 5778 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( # `  ( 1st `  u ) )  =  ( # `  ( 1st `  <. f ,  p >. ) ) )
3130eqeq1d 2384 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( ( # `  ( 1st `  u
) )  =  N  <-> 
( # `  ( 1st `  <. f ,  p >. ) )  =  N ) )
32 fveq2 5774 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. f ,  p >.  ->  ( 2nd `  u
)  =  ( 2nd `  <. f ,  p >. ) )
3332fveq1d 5776 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( ( 2nd `  u ) `  0
)  =  ( ( 2nd `  <. f ,  p >. ) `  0
) )
3433eqeq1d 2384 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( ( ( 2nd `  u ) `
 0 )  =  P  <->  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) )
3531, 34anbi12d 708 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( ( (
# `  ( 1st `  u ) )  =  N  /\  ( ( 2nd `  u ) `
 0 )  =  P )  <->  ( ( # `
 ( 1st `  <. f ,  p >. )
)  =  N  /\  ( ( 2nd `  <. f ,  p >. ) `  0 )  =  P ) ) )
3628, 35anbi12d 708 . . . . . . . . . . . . . . . . . 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 ) ) ) )
3732eqeq2d 2396 . . . . . . . . . . . . . . . . . 18  |-  ( u  =  <. f ,  p >.  ->  ( p  =  ( 2nd `  u
)  <->  p  =  ( 2nd `  <. f ,  p >. ) ) )
3836, 37anbi12d 708 . . . . . . . . . . . . . . . . 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 >. ) ) ) )
3927, 38syl5ibrcom 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
) ) ) )
4039impancom 438 . . . . . . . . . . . . . . 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
) ) ) )
4122, 40spcimedv 3118 . . . . . . . . . . . . . 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
) ) ) )
4220, 41syl5bi 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
) ) ) )
4310, 16, 42syl2anb 477 . . . . . . . . . . . 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
) ) ) )
4443exlimiv 1730 . . . . . . . . . . 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
) ) ) )
459, 44syl6bir 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 ) ) ) ) )
4645imp32 431 . . . . . . . . 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
) ) )
47 fveq2 5774 . . . . . . . . . . . . . . 15  |-  ( p  =  u  ->  ( 1st `  p )  =  ( 1st `  u
) )
4847fveq2d 5778 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  u ) ) )
4948eqeq1d 2384 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  u ) )  =  N ) )
50 fveq2 5774 . . . . . . . . . . . . . . 15  |-  ( p  =  u  ->  ( 2nd `  p )  =  ( 2nd `  u
) )
5150fveq1d 5776 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  u ) `  0
) )
5251eqeq1d 2384 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  (
( ( 2nd `  p
) `  0 )  =  P  <->  ( ( 2nd `  u ) `  0
)  =  P ) )
5349, 52anbi12d 708 . . . . . . . . . . . 12  |-  ( p  =  u  ->  (
( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P )  <->  ( ( # `
 ( 1st `  u
) )  =  N  /\  ( ( 2nd `  u ) `  0
)  =  P ) ) )
5453elrab 3182 . . . . . . . . . . 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 ) ) )
5554anbi1i 693 . . . . . . . . . 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 ) ) )
5655exbii 1675 . . . . . . . . 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
) ) )
5746, 56sylibr 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 ) ) )
58 df-rex 2738 . . . . . . . 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 ) ) )
5957, 58sylibr 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 ) )
601rexeqi 2984 . . . . . . 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
) )
6159, 60sylibr 212 . . . . . 6  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u  e.  T  p  =  ( 2nd `  u ) )
62 fveq2 5774 . . . . . . . . 9  |-  ( t  =  u  ->  ( 2nd `  t )  =  ( 2nd `  u
) )
63 fvex 5784 . . . . . . . . 9  |-  ( 2nd `  u )  e.  _V
6462, 3, 63fvmpt 5857 . . . . . . . 8  |-  ( u  e.  T  ->  ( F `  u )  =  ( 2nd `  u
) )
6564eqeq2d 2396 . . . . . . 7  |-  ( u  e.  T  ->  (
p  =  ( F `
 u )  <->  p  =  ( 2nd `  u ) ) )
6665rexbiia 2883 . . . . . 6  |-  ( E. u  e.  T  p  =  ( F `  u )  <->  E. u  e.  T  p  =  ( 2nd `  u ) )
6761, 66sylibr 212 . . . . 5  |-  ( ( V USGrph  E  /\  (
p  e.  ( ( V WWalksN  E ) `  N
)  /\  ( p `  0 )  =  P ) )  ->  E. u  e.  T  p  =  ( F `  u ) )
688, 67sylan2b 473 . . . 4  |-  ( ( V USGrph  E  /\  p  e.  W )  ->  E. u  e.  T  p  =  ( F `  u ) )
6968ralrimiva 2796 . . 3  |-  ( V USGrph  E  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
70693ad2ant1 1015 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
71 dffo3 5948 . 2  |-  ( F : T -onto-> W  <->  ( F : T --> W  /\  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) ) )
725, 70, 71sylanbrc 662 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 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034   <.cop 3950   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   -onto->wfo 5494   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   0cc0 9403   NN0cn0 10712   #chash 12307   USGrph cusg 24451   Walks cwalk 24619   WWalksN cwwlkn 24799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-usgra 24454  df-wlk 24629  df-wwlk 24800  df-wwlkn 24801
This theorem is referenced by:  wlkiswwlkbij  24841
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