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Theorem diophrex 29114
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 2449 . . . . 5  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
21rexbidv 2736 . . . 4  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  S  t  =  ( b  |`  (
1 ... N ) ) ) )
3 reseq1 5104 . . . . . 6  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
43eqeq2d 2454 . . . . 5  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
54cbvrexv 2948 . . . 4  |-  ( E. b  e.  S  t  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) )
62, 5syl6bb 261 . . 3  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) ) )
76cbvabv 2562 . 2  |-  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) }
8 eldioph3b 29103 . . . . 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 464 . . . 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 1011 . . 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 2918 . . . . . . . 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 2557 . . . . . . 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 466 . . . . . 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 2449 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
d  =  ( e  |`  ( 1 ... M
) )  <->  b  =  ( e  |`  (
1 ... M ) ) ) )
1514anbi1d 704 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1615rexbidv 2736 . . . . . . . . . . 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 3122 . . . . . . . . . 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 2873 . . . . . . . . . . . 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 1634 . . . . . . . . . . 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 2992 . . . . . . . . . . . 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 649 . . . . . . . . . . . . . . . 16  |-  ( ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
2221exbii 1634 . . . . . . . . . . . . . . 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 2975 . . . . . . . . . . . . . . . . 17  |-  e  e. 
_V
2423resex 5150 . . . . . . . . . . . . . . . 16  |-  ( e  |`  ( 1 ... M
) )  e.  _V
25 reseq1 5104 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
b  |`  ( 1 ... N ) )  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
2625eqeq2d 2454 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2726anbi2d 703 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
( ( c `  e )  =  0  /\  a  =  ( b  |`  ( 1 ... N ) ) )  <->  ( ( c `
 e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) ) )
2824, 27ceqsexv 3009 . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . 15  |-  ( ( ( c `  e
)  =  0  /\  a  =  ( ( e  |`  ( 1 ... M ) )  |`  ( 1 ... N
) ) )  <->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) )
31 simpl2 992 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  M  e.  (
ZZ>= `  N ) )
32 fzss2 11498 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... M
) )
33 resabs1 5139 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( e  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( e  |`  ( 1 ... N ) ) )
3431, 32, 333syl 20 . . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . 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 704 . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 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 2736 . . . . . . . . . . . 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 259 . . . . . . . . . . 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 259 . . . . . . . . . 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 257 . . . . . . . . 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 2557 . . . . . . . 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 29104 . . . . . . . . 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 1150 . . . . . . . 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 2517 . . . . . . 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 465 . . . . . 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 2517 . . . . 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 434 . . . 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 2841 . . 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 2528 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429   E.wrex 2716    C_ wss 3328    |` cres 4842   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   0cc0 9282   1c1 9283   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437  mzPolycmzp 29058  Diophcdioph 29093
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-hash 12104  df-mzpcl 29059  df-mzp 29060  df-dioph 29094
This theorem is referenced by:  rexrabdioph  29132
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