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Theorem 3rexfrabdioph 26857
Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1  |-  M  =  ( N  +  1 )
rexfrabdioph.2  |-  L  =  ( M  +  1 )
rexfrabdioph.3  |-  K  =  ( L  +  1 )
Assertion
Ref Expression
3rexfrabdioph  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N ) )
Distinct variable groups:    u, t,
v, w, x, K   
t, L, u, v, w, x    t, M, u, v, w, x   
t, N, u, v, w, x    ph, t
Allowed substitution hints:    ph( x, w, v, u)

Proof of Theorem 3rexfrabdioph
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fvex 5742 . . . . . . . 8  |-  ( a `
 M )  e. 
_V
2 sbc2rexg 26844 . . . . . . . 8  |-  ( ( a `  M )  e.  _V  ->  ( [. ( a `  M
)  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e. 
NN0  [. ( a `  M )  /  v ]. ph ) )
31, 2ax-mp 8 . . . . . . 7  |-  ( [. ( a `  M
)  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e. 
NN0  [. ( a `  M )  /  v ]. ph )
43sbcbii 3216 . . . . . 6  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. E. w  e.  NN0  E. x  e.  NN0  [. (
a `  M )  /  v ]. ph )
5 vex 2959 . . . . . . . 8  |-  a  e. 
_V
65resex 5186 . . . . . . 7  |-  ( a  |`  ( 1 ... N
) )  e.  _V
7 sbc2rexg 26844 . . . . . . 7  |-  ( ( a  |`  ( 1 ... N ) )  e.  _V  ->  ( [. ( a  |`  (
1 ... N ) )  /  u ]. E. w  e.  NN0  E. x  e.  NN0  [. ( a `  M )  /  v ]. ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph ) )
86, 7ax-mp 8 . . . . . 6  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. E. w  e.  NN0  E. x  e.  NN0  [. ( a `  M )  /  v ]. ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph )
94, 8bitri 241 . . . . 5  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph )
109a1i 11 . . . 4  |-  ( a  e.  ( NN0  ^m  ( 1 ... M
) )  ->  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph ) )
1110rabbiia 2946 . . 3  |-  { a  e.  ( NN0  ^m  ( 1 ... M
) )  |  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph }  =  {
a  e.  ( NN0 
^m  ( 1 ... M ) )  |  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph }
12 rexfrabdioph.1 . . . . . . 7  |-  M  =  ( N  +  1 )
13 nn0p1nn 10259 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
1412, 13syl5eqel 2520 . . . . . 6  |-  ( N  e.  NN0  ->  M  e.  NN )
1514nnnn0d 10274 . . . . 5  |-  ( N  e.  NN0  ->  M  e. 
NN0 )
1615adantr 452 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  M  e.  NN0 )
17 sbcrot3 26847 . . . . . . . . . . 11  |-  ( [. ( a `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph  <->  [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( a `  M
)  /  v ]. ph )
1817sbcbii 3216 . . . . . . . . . 10  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a `  M )  /  v ]. ph )
19 sbcrot3 26847 . . . . . . . . . 10  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
a `  M )  /  v ]. ph  <->  [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
2018, 19bitri 241 . . . . . . . . 9  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
2120sbcbii 3216 . . . . . . . 8  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
22 reseq1 5140 . . . . . . . . . 10  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
2322sbccomieg 26849 . . . . . . . . 9  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph )
24 fzssp1 11095 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
2512oveq2i 6092 . . . . . . . . . . . 12  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
2624, 25sseqtr4i 3381 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... M
)
27 resabs1 5175 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
28 dfsbcq 3163 . . . . . . . . . . 11  |-  ( ( ( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) )  ->  ( [. (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
2926, 27, 28mp2b 10 . . . . . . . . . 10  |-  ( [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph )
30 vex 2959 . . . . . . . . . . . . . 14  |-  t  e. 
_V
3130resex 5186 . . . . . . . . . . . . 13  |-  ( t  |`  ( 1 ... M
) )  e.  _V
321ax-gen 1555 . . . . . . . . . . . . 13  |-  A. a
( a `  M
)  e.  _V
33 fveq1 5727 . . . . . . . . . . . . . 14  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... M
) ) `  M
) )
3433sbcco3gOLD 3306 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) )  e.  _V  /\  A. a ( a `  M )  e.  _V )  ->  ( [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph ) )
3531, 32, 34mp2an 654 . . . . . . . . . . . 12  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph )
36 elfz1end 11081 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
3714, 36sylib 189 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
38 fvres 5745 . . . . . . . . . . . . 13  |-  ( M  e.  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
) )
39 dfsbcq 3163 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
)  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph ) )
4037, 38, 393syl 19 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph ) )
4135, 40syl5bb 249 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph ) )
4241sbcbidv 3215 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
4329, 42syl5bb 249 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
4423, 43syl5bb 249 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
4521, 44syl5bbr 251 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
4645rabbidv 2948 . . . . . 6  |-  ( N  e.  NN0  ->  { t  e.  ( NN0  ^m  ( 1 ... K
) )  |  [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph }  =  { t  e.  ( NN0  ^m  ( 1 ... K ) )  |  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph } )
4746eleq1d 2502 . . . . 5  |-  ( N  e.  NN0  ->  ( { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  K )  <->  { t  e.  ( NN0  ^m  (
1 ... K ) )  |  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) ) )
4847biimpar 472 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... K ) )  |  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  K ) )
49 rexfrabdioph.2 . . . . 5  |-  L  =  ( M  +  1 )
50 rexfrabdioph.3 . . . . 5  |-  K  =  ( L  +  1 )
5149, 502rexfrabdioph 26856 . . . 4  |-  ( ( M  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  K ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  E. x  e.  NN0  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph }  e.  (Dioph `  M )
)
5216, 48, 51syl2anc 643 . . 3  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  E. x  e.  NN0  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph }  e.  (Dioph `  M )
)
5311, 52syl5eqel 2520 . 2  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  M )
)
5412rexfrabdioph 26855 . 2  |-  ( ( N  e.  NN0  /\  { a  e.  ( NN0 
^m  ( 1 ... M ) )  | 
[. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  M )
)  ->  { u  e.  ( NN0  ^m  (
1 ... N ) )  |  E. v  e. 
NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N )
)
5553, 54syldan 457 1  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709   _Vcvv 2956   [.wsbc 3161    C_ wss 3320    |` cres 4880   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   1c1 8991    + caddc 8993   NNcn 10000   NN0cn0 10221   ...cfz 11043  Diophcdioph 26813
This theorem is referenced by:  expdiophlem2  27093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619  df-mzpcl 26780  df-mzp 26781  df-dioph 26814
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