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Theorem efgredeu 16339
Description: There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
efgred.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
efgred.s  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
Assertion
Ref Expression
efgredeu  |-  ( A  e.  W  ->  E! d  e.  D  d  .~  A )
Distinct variable groups:    A, d    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w   
k, m, t, x, T    k, d, m, n, t, v, w, x, y, z, W    .~ , d, m, t, x, y, z    S, d   
m, I, n, t, v, w, x, y, z    D, d, m, t
Allowed substitution hints:    A( x, y, z, w, v, t, k, m, n)    D( x, y, z, w, v, k, n)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y, z, w, v, n, d)    I( k, d)    M( y, z, k, d)

Proof of Theorem efgredeu
Dummy variables  a 
b  c  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . 5  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . 5  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . 5  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . 5  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . . 5  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . . 5  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
71, 2, 3, 4, 5, 6efgsfo 16326 . . . 4  |-  S : dom  S -onto-> W
8 foelrn 5947 . . . 4  |-  ( ( S : dom  S -onto-> W  /\  A  e.  W
)  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
97, 8mpan 670 . . 3  |-  ( A  e.  W  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
101, 2, 3, 4, 5, 6efgsdm 16317 . . . . . . . 8  |-  ( a  e.  dom  S  <->  ( a  e.  (Word  W  \  { (/)
} )  /\  (
a `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  a ) ) ( a `  i )  e.  ran  ( T `
 ( a `  ( i  -  1 ) ) ) ) )
1110simp2bi 1004 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  e.  D )
1211adantl 466 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  e.  D
)
131, 2, 3, 4, 5, 6efgsrel 16321 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  .~  ( S `  a ) )
1413adantl 466 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  .~  ( S `  a )
)
15 breq1 4379 . . . . . . 7  |-  ( d  =  ( a ` 
0 )  ->  (
d  .~  ( S `  a )  <->  ( a `  0 )  .~  ( S `  a ) ) )
1615rspcev 3155 . . . . . 6  |-  ( ( ( a `  0
)  e.  D  /\  ( a `  0
)  .~  ( S `  a ) )  ->  E. d  e.  D  d  .~  ( S `  a ) )
1712, 14, 16syl2anc 661 . . . . 5  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  E. d  e.  D  d  .~  ( S `  a ) )
18 breq2 4380 . . . . . 6  |-  ( A  =  ( S `  a )  ->  (
d  .~  A  <->  d  .~  ( S `  a ) ) )
1918rexbidv 2816 . . . . 5  |-  ( A  =  ( S `  a )  ->  ( E. d  e.  D  d  .~  A  <->  E. d  e.  D  d  .~  ( S `  a ) ) )
2017, 19syl5ibrcom 222 . . . 4  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( A  =  ( S `  a
)  ->  E. d  e.  D  d  .~  A ) )
2120rexlimdva 2923 . . 3  |-  ( A  e.  W  ->  ( E. a  e.  dom  S  A  =  ( S `
 a )  ->  E. d  e.  D  d  .~  A ) )
229, 21mpd 15 . 2  |-  ( A  e.  W  ->  E. d  e.  D  d  .~  A )
231, 2efger 16305 . . . . . . 7  |-  .~  Er  W
2423a1i 11 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  .~  Er  W
)
25 simprl 755 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  A
)
26 simprr 756 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  c  .~  A
)
2724, 25, 26ertr4d 7206 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  c
)
281, 2, 3, 4, 5, 6efgrelex 16338 . . . . . 6  |-  ( d  .~  c  ->  E. a  e.  ( `' S " { d } ) E. b  e.  ( `' S " { c } ) ( a `
 0 )  =  ( b `  0
) )
29 fofn 5706 . . . . . . . . . . . . . 14  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
30 fniniseg 5909 . . . . . . . . . . . . . 14  |-  ( S  Fn  dom  S  -> 
( a  e.  ( `' S " { d } )  <->  ( a  e.  dom  S  /\  ( S `  a )  =  d ) ) )
317, 29, 30mp2b 10 . . . . . . . . . . . . 13  |-  ( a  e.  ( `' S " { d } )  <-> 
( a  e.  dom  S  /\  ( S `  a )  =  d ) )
3231simplbi 460 . . . . . . . . . . . 12  |-  ( a  e.  ( `' S " { d } )  ->  a  e.  dom  S )
3332ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  a  e.  dom  S )
341, 2, 3, 4, 5, 6efgsval 16318 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  =  ( a `
 ( ( # `  a )  -  1 ) ) )
3533, 34syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  a )  =  ( a `  ( (
# `  a )  -  1 ) ) )
3631simprbi 464 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { d } )  ->  ( S `  a )  =  d )
3736ad2antrl 727 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  a )  =  d )
38 simpllr 758 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( d  e.  D  /\  c  e.  D ) )
3938simpld 459 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  d  e.  D
)
4037, 39eqeltrd 2536 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  a )  e.  D
)
411, 2, 3, 4, 5, 6efgs1b 16323 . . . . . . . . . . . . . . 15  |-  ( a  e.  dom  S  -> 
( ( S `  a )  e.  D  <->  (
# `  a )  =  1 ) )
4233, 41syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( S `
 a )  e.  D  <->  ( # `  a
)  =  1 ) )
4340, 42mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( # `  a
)  =  1 )
4443oveq1d 6191 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  a )  -  1 )  =  ( 1  -  1 ) )
45 1m1e0 10477 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
4644, 45syl6eq 2506 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  a )  -  1 )  =  0 )
4746fveq2d 5779 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( a `  ( ( # `  a
)  -  1 ) )  =  ( a `
 0 ) )
4835, 37, 473eqtr3rd 2499 . . . . . . . . 9  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( a ` 
0 )  =  d )
49 fniniseg 5909 . . . . . . . . . . . . . 14  |-  ( S  Fn  dom  S  -> 
( b  e.  ( `' S " { c } )  <->  ( b  e.  dom  S  /\  ( S `  b )  =  c ) ) )
507, 29, 49mp2b 10 . . . . . . . . . . . . 13  |-  ( b  e.  ( `' S " { c } )  <-> 
( b  e.  dom  S  /\  ( S `  b )  =  c ) )
5150simplbi 460 . . . . . . . . . . . 12  |-  ( b  e.  ( `' S " { c } )  ->  b  e.  dom  S )
5251ad2antll 728 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  b  e.  dom  S )
531, 2, 3, 4, 5, 6efgsval 16318 . . . . . . . . . . 11  |-  ( b  e.  dom  S  -> 
( S `  b
)  =  ( b `
 ( ( # `  b )  -  1 ) ) )
5452, 53syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  b )  =  ( b `  ( (
# `  b )  -  1 ) ) )
5550simprbi 464 . . . . . . . . . . 11  |-  ( b  e.  ( `' S " { c } )  ->  ( S `  b )  =  c )
5655ad2antll 728 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  b )  =  c )
5738simprd 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  c  e.  D
)
5856, 57eqeltrd 2536 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  b )  e.  D
)
591, 2, 3, 4, 5, 6efgs1b 16323 . . . . . . . . . . . . . . 15  |-  ( b  e.  dom  S  -> 
( ( S `  b )  e.  D  <->  (
# `  b )  =  1 ) )
6052, 59syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( S `
 b )  e.  D  <->  ( # `  b
)  =  1 ) )
6158, 60mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( # `  b
)  =  1 )
6261oveq1d 6191 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  b )  -  1 )  =  ( 1  -  1 ) )
6362, 45syl6eq 2506 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  b )  -  1 )  =  0 )
6463fveq2d 5779 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( b `  ( ( # `  b
)  -  1 ) )  =  ( b `
 0 ) )
6554, 56, 643eqtr3rd 2499 . . . . . . . . 9  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( b ` 
0 )  =  c )
6648, 65eqeq12d 2471 . . . . . . . 8  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  <->  d  =  c ) )
6766biimpd 207 . . . . . . 7  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  ->  d  =  c ) )
6867rexlimdvva 2930 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  ( E. a  e.  ( `' S " { d } ) E. b  e.  ( `' S " { c } ) ( a `
 0 )  =  ( b `  0
)  ->  d  =  c ) )
6928, 68syl5 32 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  ( d  .~  c  ->  d  =  c ) )
7027, 69mpd 15 . . . 4  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  =  c )
7170ex 434 . . 3  |-  ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  ->  (
( d  .~  A  /\  c  .~  A )  ->  d  =  c ) )
7271ralrimivva 2890 . 2  |-  ( A  e.  W  ->  A. d  e.  D  A. c  e.  D  ( (
d  .~  A  /\  c  .~  A )  -> 
d  =  c ) )
73 breq1 4379 . . 3  |-  ( d  =  c  ->  (
d  .~  A  <->  c  .~  A ) )
7473reu4 3236 . 2  |-  ( E! d  e.  D  d  .~  A  <->  ( E. d  e.  D  d  .~  A  /\  A. d  e.  D  A. c  e.  D  ( (
d  .~  A  /\  c  .~  A )  -> 
d  =  c ) ) )
7522, 72, 74sylanbrc 664 1  |-  ( A  e.  W  ->  E! d  e.  D  d  .~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757   A.wral 2792   E.wrex 2793   E!wreu 2794   {crab 2796    \ cdif 3409   (/)c0 3721   {csn 3961   <.cop 3967   <.cotp 3969   U_ciun 4255   class class class wbr 4376    |-> cmpt 4434    _I cid 4715    X. cxp 4922   `'ccnv 4923   dom cdm 4924   ran crn 4925   "cima 4927    Fn wfn 5497   -onto->wfo 5500   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   1oc1o 6999   2oc2o 7000    Er wer 7184   0cc0 9369   1c1 9370    - cmin 9682   ...cfz 11524  ..^cfzo 11635   #chash 12190  Word cword 12309   splice csplice 12314   <"cs2 12556   ~FG cefg 16293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-ot 3970  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-2o 7007  df-oadd 7010  df-er 7187  df-ec 7189  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-n0 10667  df-z 10734  df-uz 10949  df-rp 11079  df-fz 11525  df-fzo 11636  df-hash 12191  df-word 12317  df-concat 12319  df-s1 12320  df-substr 12321  df-splice 12322  df-s2 12563  df-efg 16296
This theorem is referenced by:  efgred2  16340  frgpnabllem2  16442
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