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Theorem seqomlem1 6478
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 5541 . . 3  |-  ( a  =  (/)  ->  ( Q `
 a )  =  ( Q `  (/) ) )
2 id 19 . . . 4  |-  ( a  =  (/)  ->  a  =  (/) )
31fveq2d 5545 . . . 4  |-  ( a  =  (/)  ->  ( 2nd `  ( Q `  a
) )  =  ( 2nd `  ( Q `
 (/) ) ) )
42, 3opeq12d 3820 . . 3  |-  ( a  =  (/)  ->  <. a ,  ( 2nd `  ( Q `  a )
) >.  =  <. (/) ,  ( 2nd `  ( Q `
 (/) ) ) >.
)
51, 4eqeq12d 2310 . 2  |-  ( a  =  (/)  ->  ( ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `
 a ) )
>. 
<->  ( Q `  (/) )  = 
<. (/) ,  ( 2nd `  ( Q `  (/) ) )
>. ) )
6 fveq2 5541 . . 3  |-  ( a  =  b  ->  ( Q `  a )  =  ( Q `  b ) )
7 id 19 . . . 4  |-  ( a  =  b  ->  a  =  b )
86fveq2d 5545 . . . 4  |-  ( a  =  b  ->  ( 2nd `  ( Q `  a ) )  =  ( 2nd `  ( Q `  b )
) )
97, 8opeq12d 3820 . . 3  |-  ( a  =  b  ->  <. a ,  ( 2nd `  ( Q `  a )
) >.  =  <. b ,  ( 2nd `  ( Q `  b )
) >. )
106, 9eqeq12d 2310 . 2  |-  ( a  =  b  ->  (
( Q `  a
)  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  b
)  =  <. b ,  ( 2nd `  ( Q `  b )
) >. ) )
11 fveq2 5541 . . 3  |-  ( a  =  suc  b  -> 
( Q `  a
)  =  ( Q `
 suc  b )
)
12 id 19 . . . 4  |-  ( a  =  suc  b  -> 
a  =  suc  b
)
1311fveq2d 5545 . . . 4  |-  ( a  =  suc  b  -> 
( 2nd `  ( Q `  a )
)  =  ( 2nd `  ( Q `  suc  b ) ) )
1412, 13opeq12d 3820 . . 3  |-  ( a  =  suc  b  ->  <. a ,  ( 2nd `  ( Q `  a
) ) >.  =  <. suc  b ,  ( 2nd `  ( Q `  suc  b ) ) >.
)
1511, 14eqeq12d 2310 . 2  |-  ( a  =  suc  b  -> 
( ( Q `  a )  =  <. a ,  ( 2nd `  ( Q `  a )
) >. 
<->  ( Q `  suc  b )  =  <. suc  b ,  ( 2nd `  ( Q `  suc  b ) ) >.
) )
16 fveq2 5541 . . 3  |-  ( a  =  A  ->  ( Q `  a )  =  ( Q `  A ) )
17 id 19 . . . 4  |-  ( a  =  A  ->  a  =  A )
1816fveq2d 5545 . . . 4  |-  ( a  =  A  ->  ( 2nd `  ( Q `  a ) )  =  ( 2nd `  ( Q `  A )
) )
1917, 18opeq12d 3820 . . 3  |-  ( a  =  A  ->  <. a ,  ( 2nd `  ( Q `  a )
) >.  =  <. A , 
( 2nd `  ( Q `  A )
) >. )
2016, 19eqeq12d 2310 . 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 5542 . . . 4  |-  ( Q `
 (/) )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I 
`  I ) >.
) `  (/) )
23 opex 4253 . . . . 5  |-  <. (/) ,  (  _I  `  I )
>.  e.  _V
2423rdg0 6450 . . . 4  |-  ( rec ( ( i  e. 
om ,  v  e. 
_V  |->  <. suc  i , 
( i F v ) >. ) ,  <. (/)
,  (  _I  `  I ) >. ) `  (/) )  =  <. (/)
,  (  _I  `  I ) >.
2522, 24eqtri 2316 . . 3  |-  ( Q `
 (/) )  =  <. (/)
,  (  _I  `  I ) >.
26 0ex 4166 . . . . . . 7  |-  (/)  e.  _V
27 fvex 5555 . . . . . . 7  |-  (  _I 
`  I )  e. 
_V
2826, 27op2nd 6145 . . . . . 6  |-  ( 2nd `  <. (/) ,  (  _I 
`  I ) >.
)  =  (  _I 
`  I )
2928eqcomi 2300 . . . . 5  |-  (  _I 
`  I )  =  ( 2nd `  <. (/)
,  (  _I  `  I ) >. )
3029opeq2i 3816 . . . 4  |-  <. (/) ,  (  _I  `  I )
>.  =  <. (/) ,  ( 2nd `  <. (/) ,  (  _I  `  I )
>. ) >.
31 id 19 . . . 4  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  ->  ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.
)
32 fveq2 5541 . . . . 5  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  ->  ( 2nd `  ( Q `  (/) ) )  =  ( 2nd `  <. (/)
,  (  _I  `  I ) >. )
)
3332opeq2d 3819 . . . 4  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  -> 
<. (/) ,  ( 2nd `  ( Q `  (/) ) )
>.  =  <. (/) ,  ( 2nd `  <. (/) ,  (  _I  `  I )
>. ) >. )
3430, 31, 333eqtr4a 2354 . . 3  |-  ( ( Q `  (/) )  = 
<. (/) ,  (  _I 
`  I ) >.  ->  ( Q `  (/) )  = 
<. (/) ,  ( 2nd `  ( Q `  (/) ) )
>. )
3525, 34ax-mp 8 . 2  |-  ( Q `
 (/) )  =  <. (/)
,  ( 2nd `  ( Q `  (/) ) )
>.
36 df-ov 5877 . . . . . 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 5555 . . . . . . 7  |-  ( 2nd `  ( Q `  b
) )  e.  _V
38 suceq 4473 . . . . . . . . 9  |-  ( i  =  b  ->  suc  i  =  suc  b )
39 oveq1 5881 . . . . . . . . 9  |-  ( i  =  b  ->  (
i F v )  =  ( b F v ) )
4038, 39opeq12d 3820 . . . . . . . 8  |-  ( i  =  b  ->  <. suc  i ,  ( i F v ) >.  =  <. suc  b ,  ( b F v ) >.
)
41 oveq2 5882 . . . . . . . . 9  |-  ( v  =  ( 2nd `  ( Q `  b )
)  ->  ( b F v )  =  ( b F ( 2nd `  ( Q `
 b ) ) ) )
4241opeq2d 3819 . . . . . . . 8  |-  ( v  =  ( 2nd `  ( Q `  b )
)  ->  <. suc  b ,  ( b F v ) >.  =  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
43 eqid 2296 . . . . . . . 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 4253 . . . . . . . 8  |-  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >.  e.  _V
4540, 42, 43, 44ovmpt2 5999 . . . . . . 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 652 . . . . . 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 2342 . . . . 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 5541 . . . . . 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 2304 . . . . 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 213 . . . 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 2804 . . . . . . . . . 10  |-  b  e. 
_V
5251sucex 4618 . . . . . . . . 9  |-  suc  b  e.  _V
53 ovex 5899 . . . . . . . . 9  |-  ( b F ( 2nd `  ( Q `  b )
) )  e.  _V
5452, 53op2nd 6145 . . . . . . . 8  |-  ( 2nd `  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >. )  =  ( b F ( 2nd `  ( Q `  b
) ) )
5554eqcomi 2300 . . . . . . 7  |-  ( b F ( 2nd `  ( Q `  b )
) )  =  ( 2nd `  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
5655a1i 10 . . . . . 6  |-  ( b  e.  om  ->  (
b F ( 2nd `  ( Q `  b
) ) )  =  ( 2nd `  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >. )
)
5756opeq2d 3819 . . . . 5  |-  ( b  e.  om  ->  <. suc  b ,  ( b F ( 2nd `  ( Q `  b )
) ) >.  =  <. suc  b ,  ( 2nd `  <. suc  b , 
( b F ( 2nd `  ( Q `
 b ) ) ) >. ) >. )
58 id 19 . . . . . 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 5541 . . . . . . 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 3819 . . . . . 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 2310 . . . . 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 213 . . . 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 40 . . 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 6465 . . . . 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 4692 . . . . . . 7  |-  ( b  e.  om  ->  suc  b  e.  om )
66 fvres 5558 . . . . . . 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 15 . . . . . 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 5542 . . . . . 6  |-  ( Q `
 suc  b )  =  ( rec (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) ,  <. (/) ,  (  _I  `  I )
>. ) `  suc  b
)
6967, 68syl6eqr 2346 . . . . 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 5558 . . . . . . 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 5542 . . . . . . 7  |-  ( Q `
 b )  =  ( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. ) `  b )
7270, 71syl6eqr 2346 . . . . . 6  |-  ( b  e.  om  ->  (
( rec ( ( i  e.  om , 
v  e.  _V  |->  <. suc  i ,  ( i F v ) >.
) ,  <. (/) ,  (  _I  `  I )
>. )  |`  om ) `  b )  =  ( Q `  b ) )
7372fveq2d 5545 . . . . 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 2336 . . . 4  |-  ( b  e.  om  ->  ( Q `  suc  b )  =  ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) `  ( Q `  b
) ) )
7574fveq2d 5545 . . . . 5  |-  ( b  e.  om  ->  ( 2nd `  ( Q `  suc  b ) )  =  ( 2nd `  (
( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v )
>. ) `  ( Q `
 b ) ) ) )
7675opeq2d 3819 . . . 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 2310 . . 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 225 . 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 4698 1  |-  ( A  e.  om  ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `
 A ) )
>. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656    _I cid 4320   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   reccrdg 6438
This theorem is referenced by:  seqomlem2  6479  seqomlem4  6481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-recs 6404  df-rdg 6439
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