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Theorem relexpcnv 25133
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 2496 . . . . . 6  |-  ( i  =  0  ->  (
i  e.  NN0  <->  0  e.  NN0 ) )
21anbi1d 686 . . . . 5  |-  ( i  =  0  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( 0  e.  NN0  /\  ph )
) )
3 oveq2 6089 . . . . . . 7  |-  ( i  =  0  ->  ( R ^ r i )  =  ( R ^
r 0 ) )
43cnveqd 5048 . . . . . 6  |-  ( i  =  0  ->  `' ( R ^ r i )  =  `' ( R ^ r 0 ) )
5 oveq2 6089 . . . . . 6  |-  ( i  =  0  ->  ( `' R ^ r i )  =  ( `' R ^ r 0 ) )
64, 5eqeq12d 2450 . . . . 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 2496 . . . . . 6  |-  ( i  =  n  ->  (
i  e.  NN0  <->  n  e.  NN0 ) )
98anbi1d 686 . . . . 5  |-  ( i  =  n  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( n  e.  NN0  /\  ph )
) )
10 oveq2 6089 . . . . . . 7  |-  ( i  =  n  ->  ( R ^ r i )  =  ( R ^
r n ) )
1110cnveqd 5048 . . . . . 6  |-  ( i  =  n  ->  `' ( R ^ r i )  =  `' ( R ^ r n ) )
12 oveq2 6089 . . . . . 6  |-  ( i  =  n  ->  ( `' R ^ r i )  =  ( `' R ^ r n ) )
1311, 12eqeq12d 2450 . . . . 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 2496 . . . . . 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 6089 . . . . . . 7  |-  ( i  =  ( n  + 
1 )  ->  ( R ^ r i )  =  ( R ^
r ( n  + 
1 ) ) )
1817cnveqd 5048 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  `' ( R ^ r i )  =  `' ( R ^ r ( n  +  1 ) ) )
19 oveq2 6089 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  ( `' R ^ r i )  =  ( `' R ^ r ( n  +  1 ) ) )
2018, 19eqeq12d 2450 . . . . 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 2496 . . . . . 6  |-  ( i  =  N  ->  (
i  e.  NN0  <->  N  e.  NN0 ) )
2322anbi1d 686 . . . . 5  |-  ( i  =  N  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( N  e.  NN0  /\  ph )
) )
24 oveq2 6089 . . . . . . 7  |-  ( i  =  N  ->  ( R ^ r i )  =  ( R ^
r N ) )
2524cnveqd 5048 . . . . . 6  |-  ( i  =  N  ->  `' ( R ^ r i )  =  `' ( R ^ r N ) )
26 oveq2 6089 . . . . . 6  |-  ( i  =  N  ->  ( `' R ^ r i )  =  ( `' R ^ r N ) )
2725, 26eqeq12d 2450 . . . . 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 25129 . . . . . 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 5048 . . . . . 6  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  `' ( R ^ r 0 )  =  `' (  _I  |`  U. U. R ) )
35 cnvresid 5523 . . . . . . 7  |-  `' (  _I  |`  U. U. R
)  =  (  _I  |`  U. U. R )
3629adantl 453 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  Rel  R )
37 relcnvfld 5400 . . . . . . . . 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 5141 . . . . . . . . 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 5242 . . . . . . . . . . 11  |-  Rel  `' R
42 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  Rel  `' R
)
43 simpl 444 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Rel  `' R
)  ->  ph )
44 cnvexg 5405 . . . . . . . . . . . . 13  |-  ( R  e.  _V  ->  `' R  e.  _V )
4543, 30, 443syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  `' R  e.  _V )
4642, 45relexp0 25129 . . . . . . . . . . 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 2441 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. `' R )  =  ( `' R ^ r 0 ) )
4939, 48sylan9eq 2488 . . . . . . . 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 2480 . . . . . 6  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' (  _I  |`  U. U. R
)  =  ( `' R ^ r 0 ) )
5234, 51sylan9eq 2488 . . . . 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 25132 . . . . . . . . . 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 5048 . . . . . . . 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 5056 . . . . . . . . . . 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 5034 . . . . . . . . . . 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 2480 . . . . . . . . . 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 2488 . . . . . . . . 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 25130 . . . . . . . . . 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 2471 . . . . . . . 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 10366 . . 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 1652    e. wcel 1725   _Vcvv 2956   U.cuni 4015    _I cid 4493   `'ccnv 4877    |` cres 4880    o. ccom 4882   Rel wrel 4883  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993   NN0cn0 10221   ^ rcrelexp 25127
This theorem is referenced by:  relexprel  25134
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  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-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-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-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  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-seq 11324  df-relexp 25128
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