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Theorem seqomlem1 6708
Description: Lemma for seq𝜔. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
Hypothesis
Ref Expression
seqomlem.a  |-  Q  =  rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)
Assertion
Ref Expression
seqomlem1  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
Distinct variable groups:    Q, i,
v    A, i, v    i, F, v
Allowed substitution hints:    I( v, i)

Proof of Theorem seqomlem1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5729 . . 3  |-  ( a  =  (/)  ->  ( Q `
 a )  =  ( Q `  (/) ) )
2 id 21 . . . 4  |-  ( a  =  (/)  ->  a  =  (/) )
31fveq2d 5733 . . . 4  |-  ( a  =  (/)  ->  ( 2nd `  ( Q `  a
) )  =  ( 2nd `  ( Q `
 (/) ) ) )
42, 3opeq12d 3993 . . 3  |-  ( a  =  (/)  ->  <. a ,  ( 2nd `  ( Q `  a )
) >.  =  <. (/) ,  ( 2nd `  ( Q `
 (/) ) ) >.
)
51, 4eqeq12d 2451 . 2  |-  ( a  =  (/)  ->  ( ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. 
<->  ( Q `  (/) )  = 
<. (/) ,  ( 2nd `  ( Q `  (/) ) )
>. ) )
6 fveq2 5729 . . 3  |-  ( a  =  b  ->  ( Q `  a )  =  ( Q `  b ) )
7 id 21 . . . 4  |-  ( a  =  b  ->  a  =  b )
86fveq2d 5733 . . . 4  |-  ( a  =  b  ->  ( 2nd `  ( Q `  a ) )  =  ( 2nd `  ( Q `  b )
) )
97, 8opeq12d 3993 . . 3  |-  ( a  =  b  ->  <. a ,  ( 2nd `  ( Q `  a )
) >.  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
106, 9eqeq12d 2451 . 2  |-  ( a  =  b  ->  (
( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  b
)  =  <. b ,  ( 2nd `  ( Q `  b )
) >. ) )
11 fveq2 5729 . . 3  |-  ( a  =  suc  b  -> 
( Q `  a
)  =  ( Q `
 suc  b )
)
12 id 21 . . . 4  |-  ( a  =  suc  b  -> 
a  =  suc  b
)
1311fveq2d 5733 . . . 4  |-  ( a  =  suc  b  -> 
( 2nd `  ( Q `  a )
)  =  ( 2nd `  ( Q `  suc  b ) ) )
1412, 13opeq12d 3993 . . 3  |-  ( a  =  suc  b  ->  <. a ,  ( 2nd `  ( Q `  a
) ) >.  =  <. suc  b ,  ( 2nd `  ( Q `  suc  b ) ) >.
)
1511, 14eqeq12d 2451 . 2  |-  ( a  =  suc  b  -> 
( ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  suc  b )  =  <. suc  b ,  ( 2nd `  ( Q `  suc  b ) ) >.
) )
16 fveq2 5729 . . 3  |-  ( a  =  A  ->  ( Q `  a )  =  ( Q `  A ) )
17 id 21 . . . 4  |-  ( a  =  A  ->  a  =  A )
1816fveq2d 5733 . . . 4  |-  ( a  =  A  ->  ( 2nd `  ( Q `  a ) )  =  ( 2nd `  ( Q `  A )
) )
1917, 18opeq12d 3993 . . 3  |-  ( a  =  A  ->  <. a ,  ( 2nd `  ( Q `  a )
) >.  =  <. A , 
( 2nd `  ( Q `  A )
) >. )
2016, 19eqeq12d 2451 . 2  |-  ( a  =  A  ->  (
( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  A
)  =  <. A , 
( 2nd `  ( Q `  A )
) >. ) )
21 seqomlem.a . . . . 5  |-  Q  =  rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)
2221fveq1i 5730 . . . 4  |-  ( Q `
 (/) )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
) `  (/) )
23 opex 4428 . . . . 5  |-  <. (/) ,  (  _I  `  I )
>.  e.  _V
2423rdg0 6680 . . . 4  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. ) `  (/) )  =  <. (/)
,  (  _I  `  I ) >.
2522, 24eqtri 2457 . . 3  |-  ( Q `
 (/) )  =  <. (/)
,  (  _I  `  I ) >.
26 0ex 4340 . . . . . . 7  |-  (/)  e.  _V
27 fvex 5743 . . . . . . 7  |-  (  _I 
`  I )  e. 
_V
2826, 27op2nd 6357 . . . . . 6  |-  ( 2nd `  <. (/) ,  (  _I 
`  I ) >.
)  =  (  _I 
`  I )
2928eqcomi 2441 . . . . 5  |-  (  _I 
`  I )  =  ( 2nd `  <. (/)
,  (  _I  `  I ) >. )
3029opeq2i 3989 . . . 4  |-  <. (/) ,  (  _I  `  I )
>.  =  <. (/) ,  ( 2nd `  <. (/) ,  (  _I  `  I )
>. ) >.
31 id 21 . . . 4  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  ->  ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.
)
32 fveq2 5729 . . . . 5  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  ->  ( 2nd `  ( Q `  (/) ) )  =  ( 2nd `  <. (/)
,  (  _I  `  I ) >. )
)
3332opeq2d 3992 . . . 4  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  -> 
<. (/) ,  ( 2nd `  ( Q `  (/) ) )
>.  =  <. (/) ,  ( 2nd `  <. (/) ,  (  _I  `  I )
>. ) >. )
3430, 31, 333eqtr4a 2495 . . 3  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  ->  ( Q `  (/) )  = 
<. (/) ,  ( 2nd `  ( Q `  (/) ) )
>. )
3525, 34ax-mp 8 . 2  |-  ( Q `
 (/) )  =  <. (/)
,  ( 2nd `  ( Q `  (/) ) )
>.
36 df-ov 6085 . . . . . 6  |-  ( b ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ( 2nd `  ( Q `  b )
) )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  <. b ,  ( 2nd `  ( Q `  b )
) >. )
37 fvex 5743 . . . . . . 7  |-  ( 2nd `  ( Q `  b
) )  e.  _V
38 suceq 4647 . . . . . . . . 9  |-  ( i  =  b  ->  suc  i  =  suc  b )
39 oveq1 6089 . . . . . . . . 9  |-  ( i  =  b  ->  (
i F v )  =  ( b F v ) )
4038, 39opeq12d 3993 . . . . . . . 8  |-  ( i  =  b  ->  <. suc  i ,  ( i F v ) >.  =  <. suc  b ,  ( b F v ) >.
)
41 oveq2 6090 . . . . . . . . 9  |-  ( v  =  ( 2nd `  ( Q `  b )
)  ->  ( b F v )  =  ( b F ( 2nd `  ( Q `
 b ) ) ) )
4241opeq2d 3992 . . . . . . . 8  |-  ( v  =  ( 2nd `  ( Q `  b )
)  ->  <. suc  b ,  ( b F v ) >.  =  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
43 eqid 2437 . . . . . . . 8  |-  ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. )  =  ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. )
44 opex 4428 . . . . . . . 8  |-  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >.  e.  _V
4540, 42, 43, 44ovmpt2 6210 . . . . . . 7  |-  ( ( b  e.  om  /\  ( 2nd `  ( Q `
 b ) )  e.  _V )  -> 
( b ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. )
( 2nd `  ( Q `  b )
) )  =  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
4637, 45mpan2 654 . . . . . 6  |-  ( b  e.  om  ->  (
b ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ( 2nd `  ( Q `  b
) ) )  = 
<. suc  b ,  ( b F ( 2nd `  ( Q `  b
) ) ) >.
)
4736, 46syl5eqr 2483 . . . . 5  |-  ( b  e.  om  ->  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  <. b ,  ( 2nd `  ( Q `  b )
) >. )  =  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
48 fveq2 5729 . . . . . 6  |-  ( ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>.  ->  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  b
) )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  <. b ,  ( 2nd `  ( Q `  b )
) >. ) )
4948eqeq1d 2445 . . . . 5  |-  ( ( Q `  b )  =  <. b ,  ( 2nd `  ( Q `
 b ) )
>.  ->  ( ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) `  ( Q `  b ) )  = 
<. suc  b ,  ( b F ( 2nd `  ( Q `  b
) ) ) >.  <->  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  <. b ,  ( 2nd `  ( Q `  b )
) >. )  =  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
)
5047, 49syl5ibrcom 215 . . . 4  |-  ( b  e.  om  ->  (
( Q `  b
)  =  <. b ,  ( 2nd `  ( Q `  b )
) >.  ->  ( (
i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) `  ( Q `  b ) )  = 
<. suc  b ,  ( b F ( 2nd `  ( Q `  b
) ) ) >.
) )
51 vex 2960 . . . . . . . . . 10  |-  b  e. 
_V
5251sucex 4792 . . . . . . . . 9  |-  suc  b  e.  _V
53 ovex 6107 . . . . . . . . 9  |-  ( b F ( 2nd `  ( Q `  b )
) )  e.  _V
5452, 53op2nd 6357 . . . . . . . 8  |-  ( 2nd `  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >. )  =  ( b F ( 2nd `  ( Q `  b
) ) )
5554eqcomi 2441 . . . . . . 7  |-  ( b F ( 2nd `  ( Q `  b )
) )  =  ( 2nd `  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
5655a1i 11 . . . . . 6  |-  ( b  e.  om  ->  (
b F ( 2nd `  ( Q `  b
) ) )  =  ( 2nd `  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
)
5756opeq2d 3992 . . . . 5  |-  ( b  e.  om  ->  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >.  =  <. suc  b ,  ( 2nd `  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >. ) >. )
58 id 21 . . . . . 6  |-  ( ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >.  ->  ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) `  ( Q `  b ) )  = 
<. suc  b ,  ( b F ( 2nd `  ( Q `  b
) ) ) >.
)
59 fveq2 5729 . . . . . . 7  |-  ( ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >.  ->  ( 2nd `  ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) `  ( Q `  b )
) )  =  ( 2nd `  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
)
6059opeq2d 3992 . . . . . 6  |-  ( ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >.  ->  <. suc  b ,  ( 2nd `  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) ) ) >.  =  <. suc  b ,  ( 2nd `  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >. ) >. )
6158, 60eqeq12d 2451 . . . . 5  |-  ( ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >.  ->  ( ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( 2nd `  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) ) ) >.  <->  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >.  =  <. suc  b ,  ( 2nd `  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >. ) >. )
)
6257, 61syl5ibrcom 215 . . . 4  |-  ( b  e.  om  ->  (
( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) `  ( Q `  b )
)  =  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >.  ->  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( 2nd `  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) ) ) >. ) )
6350, 62syld 43 . . 3  |-  ( b  e.  om  ->  (
( Q `  b
)  =  <. b ,  ( 2nd `  ( Q `  b )
) >.  ->  ( (
i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) `  ( Q `  b ) )  = 
<. suc  b ,  ( 2nd `  ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) `  ( Q `  b ) ) )
>. ) )
64 frsuc 6695 . . . . 5  |-  ( b  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  suc  b )  =  ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) `  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  b ) ) )
65 peano2 4866 . . . . . . 7  |-  ( b  e.  om  ->  suc  b  e.  om )
66 fvres 5746 . . . . . . 7  |-  ( suc  b  e.  om  ->  ( ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  suc  b )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  b
) )
6765, 66syl 16 . . . . . 6  |-  ( b  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  suc  b )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  b
) )
6821fveq1i 5730 . . . . . 6  |-  ( Q `
 suc  b )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  b
)
6967, 68syl6eqr 2487 . . . . 5  |-  ( b  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  suc  b )  =  ( Q `  suc  b ) )
70 fvres 5746 . . . . . . 7  |-  ( b  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  b )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
) `  b )
)
7121fveq1i 5730 . . . . . . 7  |-  ( Q `
 b )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  b )
7270, 71syl6eqr 2487 . . . . . 6  |-  ( b  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  b )  =  ( Q `  b ) )
7372fveq2d 5733 . . . . 5  |-  ( b  e.  om  ->  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
)  |`  om ) `  b ) )  =  ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) `  ( Q `  b )
) )
7464, 69, 733eqtr3d 2477 . . . 4  |-  ( b  e.  om  ->  ( Q `  suc  b )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  b
) ) )
7574fveq2d 5733 . . . . 5  |-  ( b  e.  om  ->  ( 2nd `  ( Q `  suc  b ) )  =  ( 2nd `  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) ) ) )
7675opeq2d 3992 . . . 4  |-  ( b  e.  om  ->  <. suc  b ,  ( 2nd `  ( Q `  suc  b ) ) >.  =  <. suc  b ,  ( 2nd `  ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) `  ( Q `  b )
) ) >. )
7774, 76eqeq12d 2451 . . 3  |-  ( b  e.  om  ->  (
( Q `  suc  b )  =  <. suc  b ,  ( 2nd `  ( Q `  suc  b ) ) >.  <->  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) )  =  <. suc  b , 
( 2nd `  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) ) ) >. ) )
7863, 77sylibrd 227 . 2  |-  ( b  e.  om  ->  (
( Q `  b
)  =  <. b ,  ( 2nd `  ( Q `  b )
) >.  ->  ( Q `  suc  b )  = 
<. suc  b ,  ( 2nd `  ( Q `
 suc  b )
) >. ) )
795, 10, 15, 20, 35, 78finds 4872 1  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2957   (/)c0 3629   <.cop 3818    _I cid 4494   suc csuc 4584   omcom 4846    |` cres 4881   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   2ndc2nd 6349   reccrdg 6668
This theorem is referenced by:  seqomlem2  6709  seqomlem4  6711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-recs 6634  df-rdg 6669
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