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Theorem relexpcnv 25086
Description: Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
relexpcnv.1  |-  ( ph  ->  Rel  R )
relexpcnv.2  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
relexpcnv  |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) )

Proof of Theorem relexpcnv
Dummy variables  i  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2464 . . . . . 6  |-  ( i  =  0  ->  (
i  e.  NN0  <->  0  e.  NN0 ) )
21anbi1d 686 . . . . 5  |-  ( i  =  0  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( 0  e.  NN0  /\  ph )
) )
3 oveq2 6048 . . . . . . 7  |-  ( i  =  0  ->  ( R ^ r i )  =  ( R ^
r 0 ) )
43cnveqd 5007 . . . . . 6  |-  ( i  =  0  ->  `' ( R ^ r i )  =  `' ( R ^ r 0 ) )
5 oveq2 6048 . . . . . 6  |-  ( i  =  0  ->  ( `' R ^ r i )  =  ( `' R ^ r 0 ) )
64, 5eqeq12d 2418 . . . . 5  |-  ( i  =  0  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) ) )
72, 6imbi12d 312 . . . 4  |-  ( i  =  0  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( 0  e. 
NN0  /\  ph )  ->  `' ( R ^
r 0 )  =  ( `' R ^
r 0 ) ) ) )
8 eleq1 2464 . . . . . 6  |-  ( i  =  n  ->  (
i  e.  NN0  <->  n  e.  NN0 ) )
98anbi1d 686 . . . . 5  |-  ( i  =  n  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( n  e.  NN0  /\  ph )
) )
10 oveq2 6048 . . . . . . 7  |-  ( i  =  n  ->  ( R ^ r i )  =  ( R ^
r n ) )
1110cnveqd 5007 . . . . . 6  |-  ( i  =  n  ->  `' ( R ^ r i )  =  `' ( R ^ r n ) )
12 oveq2 6048 . . . . . 6  |-  ( i  =  n  ->  ( `' R ^ r i )  =  ( `' R ^ r n ) )
1311, 12eqeq12d 2418 . . . . 5  |-  ( i  =  n  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r n )  =  ( `' R ^ r n ) ) )
149, 13imbi12d 312 . . . 4  |-  ( i  =  n  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( n  e. 
NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) ) ) )
15 eleq1 2464 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  (
i  e.  NN0  <->  ( n  +  1 )  e. 
NN0 ) )
1615anbi1d 686 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( (
n  +  1 )  e.  NN0  /\  ph )
) )
17 oveq2 6048 . . . . . . 7  |-  ( i  =  ( n  + 
1 )  ->  ( R ^ r i )  =  ( R ^
r ( n  + 
1 ) ) )
1817cnveqd 5007 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  `' ( R ^ r i )  =  `' ( R ^ r ( n  +  1 ) ) )
19 oveq2 6048 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  ( `' R ^ r i )  =  ( `' R ^ r ( n  +  1 ) ) )
2018, 19eqeq12d 2418 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) ) )
2116, 20imbi12d 312 . . . 4  |-  ( i  =  ( n  + 
1 )  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( ( n  +  1 )  e. 
NN0  /\  ph )  ->  `' ( R ^
r ( n  + 
1 ) )  =  ( `' R ^
r ( n  + 
1 ) ) ) ) )
22 eleq1 2464 . . . . . 6  |-  ( i  =  N  ->  (
i  e.  NN0  <->  N  e.  NN0 ) )
2322anbi1d 686 . . . . 5  |-  ( i  =  N  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( N  e.  NN0  /\  ph )
) )
24 oveq2 6048 . . . . . . 7  |-  ( i  =  N  ->  ( R ^ r i )  =  ( R ^
r N ) )
2524cnveqd 5007 . . . . . 6  |-  ( i  =  N  ->  `' ( R ^ r i )  =  `' ( R ^ r N ) )
26 oveq2 6048 . . . . . 6  |-  ( i  =  N  ->  ( `' R ^ r i )  =  ( `' R ^ r N ) )
2725, 26eqeq12d 2418 . . . . 5  |-  ( i  =  N  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
2823, 27imbi12d 312 . . . 4  |-  ( i  =  N  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( N  e. 
NN0  /\  ph )  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) ) )
29 relexpcnv.1 . . . . . . 7  |-  ( ph  ->  Rel  R )
30 relexpcnv.2 . . . . . . 7  |-  ( ph  ->  R  e.  _V )
3129, 30relexp0 25082 . . . . . 6  |-  ( ph  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R
) )
3231adantl 453 . . . . 5  |-  ( ( 0  e.  NN0  /\  ph )  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R ) )
33 id 20 . . . . . . 7  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  ( R ^
r 0 )  =  (  _I  |`  U. U. R ) )
3433cnveqd 5007 . . . . . 6  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  `' ( R ^ r 0 )  =  `' (  _I  |`  U. U. R ) )
35 cnvresid 5482 . . . . . . 7  |-  `' (  _I  |`  U. U. R
)  =  (  _I  |`  U. U. R )
3629adantl 453 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  Rel  R )
37 relcnvfld 5359 . . . . . . . . 9  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
3836, 37syl 16 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  ph )  ->  U. U. R  =  U. U. `' R
)
39 reseq2 5100 . . . . . . . . 9  |-  ( U. U. R  =  U. U. `' R  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
40 simpr 448 . . . . . . . . . . 11  |-  ( ( 0  e.  NN0  /\  ph )  ->  ph )
41 relcnv 5201 . . . . . . . . . . 11  |-  Rel  `' R
42 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  Rel  `' R
)
43 simpl 444 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Rel  `' R
)  ->  ph )
44 cnvexg 5364 . . . . . . . . . . . . 13  |-  ( R  e.  _V  ->  `' R  e.  _V )
4543, 30, 443syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  `' R  e.  _V )
4642, 45relexp0 25082 . . . . . . . . . . 11  |-  ( (
ph  /\  Rel  `' R
)  ->  ( `' R ^ r 0 )  =  (  _I  |`  U. U. `' R ) )
4740, 41, 46sylancl 644 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  ph )  ->  ( `' R ^ r 0 )  =  (  _I  |`  U. U. `' R ) )
4847eqcomd 2409 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. `' R )  =  ( `' R ^ r 0 ) )
4939, 48sylan9eq 2456 . . . . . . . 8  |-  ( ( U. U. R  = 
U. U. `' R  /\  ( 0  e.  NN0  /\ 
ph ) )  -> 
(  _I  |`  U. U. R )  =  ( `' R ^ r 0 ) )
5038, 49mpancom 651 . . . . . . 7  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. R )  =  ( `' R ^
r 0 ) )
5135, 50syl5eq 2448 . . . . . 6  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' (  _I  |`  U. U. R
)  =  ( `' R ^ r 0 ) )
5234, 51sylan9eq 2456 . . . . 5  |-  ( ( ( R ^ r 0 )  =  (  _I  |`  U. U. R
)  /\  ( 0  e.  NN0  /\  ph )
)  ->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) )
5332, 52mpancom 651 . . . 4  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) )
54 simprrr 742 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  n  e.  NN0 )
55 simpl 444 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) )  ->  ph )
5655adantl 453 . . . . . . . . . . . 12  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ph )
5756, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  Rel  R )
5856, 30syl 16 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  R  e.  _V )
5957, 58relexpsucl 25085 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( n  e. 
NN0  ->  ( R ^
r ( n  + 
1 ) )  =  ( R  o.  ( R ^ r n ) ) ) )
6054, 59mpd 15 . . . . . . . . 9  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( R ^
r ( n  + 
1 ) )  =  ( R  o.  ( R ^ r n ) ) )
6160cnveqd 5007 . . . . . . . 8  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^
r n ) ) )
62 id 20 . . . . . . . . . 10  |-  ( `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^ r n ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^
r n ) ) )
63 cnvco 5015 . . . . . . . . . . 11  |-  `' ( R  o.  ( R ^ r n ) )  =  ( `' ( R ^ r n )  o.  `' R )
64 simprrl 741 . . . . . . . . . . . . 13  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( ( n  e.  NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) ) )
6554, 56, 64mp2and 661 . . . . . . . . . . . 12  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )
6665coeq1d 4993 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( `' ( R ^ r n )  o.  `' R
)  =  ( ( `' R ^ r n )  o.  `' R
) )
6763, 66syl5eq 2448 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R  o.  ( R ^
r n ) )  =  ( ( `' R ^ r n )  o.  `' R
) )
6862, 67sylan9eq 2456 . . . . . . . . 9  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( ( `' R ^ r n )  o.  `' R
) )
6954adantl 453 . . . . . . . . . 10  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  n  e.  NN0 )
7041a1i 11 . . . . . . . . . . 11  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  Rel  `' R
)
71 simprrl 741 . . . . . . . . . . . 12  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  ph )
7271, 30, 443syl 19 . . . . . . . . . . 11  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  `' R  e.  _V )
7370, 72relexpsucr 25083 . . . . . . . . . 10  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  ( n  e.  NN0  ->  ( `' R ^ r ( n  +  1 ) )  =  ( ( `' R ^ r n )  o.  `' R
) ) )
7469, 73mpd 15 . . . . . . . . 9  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  ( `' R ^ r ( n  +  1 ) )  =  ( ( `' R ^ r n )  o.  `' R
) )
7568, 74eqtr4d 2439 . . . . . . . 8  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) )
7661, 75mpancom 651 . . . . . . 7  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) )
7776anassrs 630 . . . . . 6  |-  ( ( ( ( n  + 
1 )  e.  NN0  /\ 
ph )  /\  (
( ( n  e. 
NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) )  /\  n  e.  NN0 ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) )
7877expcom 425 . . . . 5  |-  ( ( ( ( n  e. 
NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) )  /\  n  e.  NN0 )  ->  ( ( ( n  +  1 )  e.  NN0  /\  ph )  ->  `' ( R ^
r ( n  + 
1 ) )  =  ( `' R ^
r ( n  + 
1 ) ) ) )
7978expcom 425 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  ->  (
( ( n  + 
1 )  e.  NN0  /\ 
ph )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) ) ) )
807, 14, 21, 28, 53, 79nn0ind 10322 . . 3  |-  ( N  e.  NN0  ->  ( ( N  e.  NN0  /\  ph )  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
8180anabsi5 791 . 2  |-  ( ( N  e.  NN0  /\  ph )  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) )
8281expcom 425 1  |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   U.cuni 3975    _I cid 4453   `'ccnv 4836    |` cres 4839    o. ccom 4841   Rel wrel 4842  (class class class)co 6040   0cc0 8946   1c1 8947    + caddc 8949   NN0cn0 10177   ^ rcrelexp 25080
This theorem is referenced by:  relexprel  25087
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-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  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-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-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-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  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-seq 11279  df-relexp 25081
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