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Theorem diophrex 26724
Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
diophrex  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
Distinct variable groups:    t, N, u    t, S, u
Allowed substitution hints:    M( u, t)

Proof of Theorem diophrex
Dummy variables  a 
b  c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2410 . . . . 5  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
21rexbidv 2687 . . . 4  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  S  t  =  ( b  |`  (
1 ... N ) ) ) )
3 reseq1 5099 . . . . . 6  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
43eqeq2d 2415 . . . . 5  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
54cbvrexv 2893 . . . 4  |-  ( E. b  e.  S  t  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) )
62, 5syl6bb 253 . . 3  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) ) )
76cbvabv 2523 . 2  |-  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) }
8 eldioph3b 26713 . . . . 5  |-  ( S  e.  (Dioph `  M
)  <->  ( M  e. 
NN0  /\  E. c  e.  (mzPoly `  NN ) S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } ) )
98simprbi 451 . . . 4  |-  ( S  e.  (Dioph `  M
)  ->  E. c  e.  (mzPoly `  NN ) S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } )
1093ad2ant3 980 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  E. c  e.  (mzPoly `  NN ) S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } )
11 rexeq 2865 . . . . . . . 8  |-  ( S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) }  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } a  =  ( b  |`  ( 1 ... N
) ) ) )
1211abbidv 2518 . . . . . . 7  |-  ( S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) }  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { a  |  E. b  e. 
{ d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } a  =  ( b  |`  ( 1 ... N
) ) } )
1312adantl 453 . . . . . 6  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) )  /\  c  e.  (mzPoly `  NN ) )  /\  S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } )  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  =  { a  |  E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) } )
14 eqeq1 2410 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
d  =  ( e  |`  ( 1 ... M
) )  <->  b  =  ( e  |`  (
1 ... M ) ) ) )
1514anbi1d 686 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1615rexbidv 2687 . . . . . . . . . . 11  |-  ( d  =  b  ->  ( E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  <->  E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1716rexab 3057 . . . . . . . . . 10  |-  ( E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) )  <->  E. b ( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
18 r19.41v 2821 . . . . . . . . . . . 12  |-  ( E. e  e.  ( NN0 
^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
1918exbii 1589 . . . . . . . . . . 11  |-  ( E. b E. e  e.  ( NN0  ^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. b
( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
20 rexcom4 2935 . . . . . . . . . . . 12  |-  ( E. e  e.  ( NN0 
^m  NN ) E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. b E. e  e.  ( NN0  ^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
21 anass 631 . . . . . . . . . . . . . . . 16  |-  ( ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
2221exbii 1589 . . . . . . . . . . . . . . 15  |-  ( E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. b
( b  =  ( e  |`  ( 1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
23 vex 2919 . . . . . . . . . . . . . . . . 17  |-  e  e. 
_V
2423resex 5145 . . . . . . . . . . . . . . . 16  |-  ( e  |`  ( 1 ... M
) )  e.  _V
25 reseq1 5099 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
b  |`  ( 1 ... N ) )  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
2625eqeq2d 2415 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2726anbi2d 685 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
( ( c `  e )  =  0  /\  a  =  ( b  |`  ( 1 ... N ) ) )  <->  ( ( c `
 e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) ) )
2824, 27ceqsexv 2951 . . . . . . . . . . . . . . 15  |-  ( E. b ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) )  <->  ( (
c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2922, 28bitri 241 . . . . . . . . . . . . . 14  |-  ( E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  ( (
c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
30 ancom 438 . . . . . . . . . . . . . . 15  |-  ( ( ( c `  e
)  =  0  /\  a  =  ( ( e  |`  ( 1 ... M ) )  |`  ( 1 ... N
) ) )  <->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) )
31 simpl2 961 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  M  e.  (
ZZ>= `  N ) )
32 fzss2 11048 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... M
) )
33 resabs1 5134 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( e  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( e  |`  ( 1 ... N ) ) )
3431, 32, 333syl 19 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  =  ( e  |`  ( 1 ... N
) ) )
3534eqeq2d 2415 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  <-> 
a  =  ( e  |`  ( 1 ... N
) ) ) )
3635anbi1d 686 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 )  <->  ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3730, 36syl5bb 249 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( ( c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )  <->  ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3829, 37syl5bb 249 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b
( ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3938rexbidv 2687 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. e  e.  ( NN0  ^m  NN ) E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
4020, 39syl5bbr 251 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b E. e  e.  ( NN0  ^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  ( 1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
4119, 40syl5bbr 251 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b
( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  ( 1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
4217, 41syl5bb 249 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } a  =  ( b  |`  ( 1 ... N
) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
4342abbidv 2518 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. b  e.  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) }  =  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) } )
44 eldioph3 26714 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) }  e.  (Dioph `  N ) )
45443ad2antl1 1119 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  ( 1 ... N ) )  /\  ( c `  e )  =  0 ) }  e.  (Dioph `  N ) )
4643, 45eqeltrd 2478 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. b  e.  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) }  e.  (Dioph `  N ) )
4746adantr 452 . . . . . 6  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) )  /\  c  e.  (mzPoly `  NN ) )  /\  S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } )  ->  { a  |  E. b  e.  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) }  e.  (Dioph `  N ) )
4813, 47eqeltrd 2478 . . . . 5  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) )  /\  c  e.  (mzPoly `  NN ) )  /\  S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } )  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
4948ex 424 . . . 4  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) }  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) ) )
5049rexlimdva 2790 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  ( E. c  e.  (mzPoly `  NN ) S  =  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) }  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  e.  (Dioph `  N ) ) )
5110, 50mpd 15 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
527, 51syl5eqelr 2489 1  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667    C_ wss 3280    |` cres 4839   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   0cc0 8946   1c1 8947   NNcn 9956   NN0cn0 10177   ZZ>=cuz 10444   ...cfz 10999  mzPolycmzp 26669  Diophcdioph 26703
This theorem is referenced by:  rexrabdioph  26744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-mzpcl 26670  df-mzp 26671  df-dioph 26704
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