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Theorem wwlktovf1 12847
Description: Lemma 2 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
Hypotheses
Ref Expression
wrd2f1tovbij.d  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }
wrd2f1tovbij.r  |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }
wrd2f1tovbij.f  |-  F  =  ( t  e.  D  |->  ( t `  1
) )
Assertion
Ref Expression
wwlktovf1  |-  F : D -1-1-> R
Distinct variable groups:    t, D    P, n, t, w    t, R    n, V, t, w   
n, X, w
Allowed substitution hints:    D( w, n)    R( w, n)    F( w, t, n)    X( t)

Proof of Theorem wwlktovf1
Dummy variables  x  y  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd2f1tovbij.d . . 3  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }
2 wrd2f1tovbij.r . . 3  |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }
3 wrd2f1tovbij.f . . 3  |-  F  =  ( t  e.  D  |->  ( t `  1
) )
41, 2, 3wwlktovf 12846 . 2  |-  F : D
--> R
5 fvex 5869 . . . . . 6  |-  ( x `
 1 )  e. 
_V
6 fveq1 5858 . . . . . . 7  |-  ( t  =  x  ->  (
t `  1 )  =  ( x ` 
1 ) )
76, 3fvmptg 5941 . . . . . 6  |-  ( ( x  e.  D  /\  ( x `  1
)  e.  _V )  ->  ( F `  x
)  =  ( x `
 1 ) )
85, 7mpan2 671 . . . . 5  |-  ( x  e.  D  ->  ( F `  x )  =  ( x ` 
1 ) )
9 fvex 5869 . . . . . 6  |-  ( y `
 1 )  e. 
_V
10 fveq1 5858 . . . . . . 7  |-  ( t  =  y  ->  (
t `  1 )  =  ( y ` 
1 ) )
1110, 3fvmptg 5941 . . . . . 6  |-  ( ( y  e.  D  /\  ( y `  1
)  e.  _V )  ->  ( F `  y
)  =  ( y `
 1 ) )
129, 11mpan2 671 . . . . 5  |-  ( y  e.  D  ->  ( F `  y )  =  ( y ` 
1 ) )
138, 12eqeqan12d 2485 . . . 4  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
( x `  1
)  =  ( y `
 1 ) ) )
141eleq2i 2540 . . . . . 6  |-  ( x  e.  D  <->  x  e.  { w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) } )
15 fveq2 5859 . . . . . . . . 9  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
1615eqeq1d 2464 . . . . . . . 8  |-  ( w  =  x  ->  (
( # `  w )  =  2  <->  ( # `  x
)  =  2 ) )
17 fveq1 5858 . . . . . . . . 9  |-  ( w  =  x  ->  (
w `  0 )  =  ( x ` 
0 ) )
1817eqeq1d 2464 . . . . . . . 8  |-  ( w  =  x  ->  (
( w `  0
)  =  P  <->  ( x `  0 )  =  P ) )
19 fveq1 5858 . . . . . . . . . 10  |-  ( w  =  x  ->  (
w `  1 )  =  ( x ` 
1 ) )
2017, 19preq12d 4109 . . . . . . . . 9  |-  ( w  =  x  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( x ` 
0 ) ,  ( x `  1 ) } )
2120eleq1d 2531 . . . . . . . 8  |-  ( w  =  x  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
x `  0 ) ,  ( x ` 
1 ) }  e.  X ) )
2216, 18, 213anbi123d 1294 . . . . . . 7  |-  ( w  =  x  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X )  <->  ( ( # `
 x )  =  2  /\  ( x `
 0 )  =  P  /\  { ( x `  0 ) ,  ( x ` 
1 ) }  e.  X ) ) )
2322elrab 3256 . . . . . 6  |-  ( x  e.  { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  <->  ( x  e. Word  V  /\  ( (
# `  x )  =  2  /\  (
x `  0 )  =  P  /\  { ( x `  0 ) ,  ( x ` 
1 ) }  e.  X ) ) )
2414, 23bitri 249 . . . . 5  |-  ( x  e.  D  <->  ( x  e. Word  V  /\  ( (
# `  x )  =  2  /\  (
x `  0 )  =  P  /\  { ( x `  0 ) ,  ( x ` 
1 ) }  e.  X ) ) )
251eleq2i 2540 . . . . . 6  |-  ( y  e.  D  <->  y  e.  { w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X ) } )
26 fveq2 5859 . . . . . . . . 9  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
2726eqeq1d 2464 . . . . . . . 8  |-  ( w  =  y  ->  (
( # `  w )  =  2  <->  ( # `  y
)  =  2 ) )
28 fveq1 5858 . . . . . . . . 9  |-  ( w  =  y  ->  (
w `  0 )  =  ( y ` 
0 ) )
2928eqeq1d 2464 . . . . . . . 8  |-  ( w  =  y  ->  (
( w `  0
)  =  P  <->  ( y `  0 )  =  P ) )
30 fveq1 5858 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w `  1 )  =  ( y ` 
1 ) )
3128, 30preq12d 4109 . . . . . . . . 9  |-  ( w  =  y  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( y ` 
0 ) ,  ( y `  1 ) } )
3231eleq1d 2531 . . . . . . . 8  |-  ( w  =  y  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
y `  0 ) ,  ( y ` 
1 ) }  e.  X ) )
3327, 29, 323anbi123d 1294 . . . . . . 7  |-  ( w  =  y  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X )  <->  ( ( # `
 y )  =  2  /\  ( y `
 0 )  =  P  /\  { ( y `  0 ) ,  ( y ` 
1 ) }  e.  X ) ) )
3433elrab 3256 . . . . . 6  |-  ( y  e.  { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  X ) }  <->  ( y  e. Word  V  /\  ( (
# `  y )  =  2  /\  (
y `  0 )  =  P  /\  { ( y `  0 ) ,  ( y ` 
1 ) }  e.  X ) ) )
3525, 34bitri 249 . . . . 5  |-  ( y  e.  D  <->  ( y  e. Word  V  /\  ( (
# `  y )  =  2  /\  (
y `  0 )  =  P  /\  { ( y `  0 ) ,  ( y ` 
1 ) }  e.  X ) ) )
36 simp1 991 . . . . . . . . . 10  |-  ( ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X )  -> 
( # `  x )  =  2 )
3736adantl 466 . . . . . . . . 9  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  ->  ( # `  x
)  =  2 )
38 simp1 991 . . . . . . . . . . 11  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  -> 
( # `  y )  =  2 )
3938eqcomd 2470 . . . . . . . . . 10  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  -> 
2  =  ( # `  y ) )
4039adantl 466 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) )  ->  2  =  (
# `  y )
)
4137, 40sylan9eq 2523 . . . . . . . 8  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( # `  x )  =  ( # `  y
) )
4241adantr 465 . . . . . . 7  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  ( # `
 x )  =  ( # `  y
) )
43 simp2 992 . . . . . . . . . . 11  |-  ( ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X )  -> 
( x `  0
)  =  P )
4443adantl 466 . . . . . . . . . 10  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  ->  ( x ` 
0 )  =  P )
45 simp2 992 . . . . . . . . . . . 12  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  -> 
( y `  0
)  =  P )
4645eqcomd 2470 . . . . . . . . . . 11  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  ->  P  =  ( y `  0 ) )
4746adantl 466 . . . . . . . . . 10  |-  ( ( y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) )  ->  P  =  ( y `  0 ) )
4844, 47sylan9eq 2523 . . . . . . . . 9  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( x `  0
)  =  ( y `
 0 ) )
4948adantr 465 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x `  0 )  =  ( y ` 
0 ) )
50 simpr 461 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x `  1 )  =  ( y ` 
1 ) )
51 oveq2 6285 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  2  ->  (
0..^ ( # `  x
) )  =  ( 0..^ 2 ) )
52 fzo0to2pr 11858 . . . . . . . . . . . . 13  |-  ( 0..^ 2 )  =  {
0 ,  1 }
5351, 52syl6eq 2519 . . . . . . . . . . . 12  |-  ( (
# `  x )  =  2  ->  (
0..^ ( # `  x
) )  =  {
0 ,  1 } )
5453raleqdv 3059 . . . . . . . . . . 11  |-  ( (
# `  x )  =  2  ->  ( A. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
)  <->  A. i  e.  {
0 ,  1 }  ( x `  i
)  =  ( y `
 i ) ) )
55 c0ex 9581 . . . . . . . . . . . 12  |-  0  e.  _V
56 1ex 9582 . . . . . . . . . . . 12  |-  1  e.  _V
57 fveq2 5859 . . . . . . . . . . . . . 14  |-  ( i  =  0  ->  (
x `  i )  =  ( x ` 
0 ) )
58 fveq2 5859 . . . . . . . . . . . . . 14  |-  ( i  =  0  ->  (
y `  i )  =  ( y ` 
0 ) )
5957, 58eqeq12d 2484 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
( x `  i
)  =  ( y `
 i )  <->  ( x `  0 )  =  ( y `  0
) ) )
60 fveq2 5859 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  (
x `  i )  =  ( x ` 
1 ) )
61 fveq2 5859 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  (
y `  i )  =  ( y ` 
1 ) )
6260, 61eqeq12d 2484 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
( x `  i
)  =  ( y `
 i )  <->  ( x `  1 )  =  ( y `  1
) ) )
6359, 62ralprg 4071 . . . . . . . . . . . 12  |-  ( ( 0  e.  _V  /\  1  e.  _V )  ->  ( A. i  e. 
{ 0 ,  1 }  ( x `  i )  =  ( y `  i )  <-> 
( ( x ` 
0 )  =  ( y `  0 )  /\  ( x ` 
1 )  =  ( y `  1 ) ) ) )
6455, 56, 63mp2an 672 . . . . . . . . . . 11  |-  ( A. i  e.  { 0 ,  1 }  (
x `  i )  =  ( y `  i )  <->  ( (
x `  0 )  =  ( y ` 
0 )  /\  (
x `  1 )  =  ( y ` 
1 ) ) )
6554, 64syl6bb 261 . . . . . . . . . 10  |-  ( (
# `  x )  =  2  ->  ( A. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
)  <->  ( ( x `
 0 )  =  ( y `  0
)  /\  ( x `  1 )  =  ( y `  1
) ) ) )
66653ad2ant1 1012 . . . . . . . . 9  |-  ( ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X )  -> 
( A. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =  ( y `  i )  <->  ( (
x `  0 )  =  ( y ` 
0 )  /\  (
x `  1 )  =  ( y ` 
1 ) ) ) )
6766ad3antlr 730 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  ( A. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
)  <->  ( ( x `
 0 )  =  ( y `  0
)  /\  ( x `  1 )  =  ( y `  1
) ) ) )
6849, 50, 67mpbir2and 915 . . . . . . 7  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  A. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =  ( y `  i ) )
69 simpl 457 . . . . . . . . . 10  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  ->  x  e. Word  V
)
70 simpl 457 . . . . . . . . . 10  |-  ( ( y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) )  ->  y  e. Word  V
)
7169, 70anim12i 566 . . . . . . . . 9  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( x  e. Word  V  /\  y  e. Word  V ) )
7271adantr 465 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x  e. Word  V  /\  y  e. Word  V )
)
73 eqwrd 12536 . . . . . . . 8  |-  ( ( x  e. Word  V  /\  y  e. Word  V )  ->  ( x  =  y  <-> 
( ( # `  x
)  =  ( # `  y )  /\  A. i  e.  ( 0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
) ) ) )
7472, 73syl 16 . . . . . . 7  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x  =  y  <->  ( ( # `
 x )  =  ( # `  y
)  /\  A. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =  ( y `  i ) ) ) )
7542, 68, 74mpbir2and 915 . . . . . 6  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  x  =  y )
7675ex 434 . . . . 5  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( ( x ` 
1 )  =  ( y `  1 )  ->  x  =  y ) )
7724, 35, 76syl2anb 479 . . . 4  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( x ` 
1 )  =  ( y `  1 )  ->  x  =  y ) )
7813, 77sylbid 215 . . 3  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
7978rgen2a 2886 . 2  |-  A. x  e.  D  A. y  e.  D  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
80 dff13 6147 . 2  |-  ( F : D -1-1-> R  <->  ( F : D --> R  /\  A. x  e.  D  A. y  e.  D  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
814, 79, 80mpbir2an 913 1  |-  F : D -1-1-> R
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813   _Vcvv 3108   {cpr 4024    |-> cmpt 4500   -->wf 5577   -1-1->wf1 5578   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484   2c2 10576  ..^cfzo 11783   #chash 12362  Word cword 12489
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-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-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-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  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
This theorem is referenced by:  wwlktovf1o  12849
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