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Theorem relexpcnv 24044
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 2356 . . . . . 6  |-  ( i  =  0  ->  (
i  e.  NN0  <->  0  e.  NN0 ) )
21anbi1d 685 . . . . 5  |-  ( i  =  0  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( 0  e.  NN0  /\  ph )
) )
3 oveq2 5882 . . . . . . 7  |-  ( i  =  0  ->  ( R ^ r i )  =  ( R ^
r 0 ) )
43cnveqd 4873 . . . . . 6  |-  ( i  =  0  ->  `' ( R ^ r i )  =  `' ( R ^ r 0 ) )
5 oveq2 5882 . . . . . 6  |-  ( i  =  0  ->  ( `' R ^ r i )  =  ( `' R ^ r 0 ) )
64, 5eqeq12d 2310 . . . . 5  |-  ( i  =  0  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) ) )
72, 6imbi12d 311 . . . 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 2356 . . . . . 6  |-  ( i  =  n  ->  (
i  e.  NN0  <->  n  e.  NN0 ) )
98anbi1d 685 . . . . 5  |-  ( i  =  n  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( n  e.  NN0  /\  ph )
) )
10 oveq2 5882 . . . . . . 7  |-  ( i  =  n  ->  ( R ^ r i )  =  ( R ^
r n ) )
1110cnveqd 4873 . . . . . 6  |-  ( i  =  n  ->  `' ( R ^ r i )  =  `' ( R ^ r n ) )
12 oveq2 5882 . . . . . 6  |-  ( i  =  n  ->  ( `' R ^ r i )  =  ( `' R ^ r n ) )
1311, 12eqeq12d 2310 . . . . 5  |-  ( i  =  n  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r n )  =  ( `' R ^ r n ) ) )
149, 13imbi12d 311 . . . 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 2356 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  (
i  e.  NN0  <->  ( n  +  1 )  e. 
NN0 ) )
1615anbi1d 685 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( (
n  +  1 )  e.  NN0  /\  ph )
) )
17 oveq2 5882 . . . . . . 7  |-  ( i  =  ( n  + 
1 )  ->  ( R ^ r i )  =  ( R ^
r ( n  + 
1 ) ) )
1817cnveqd 4873 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  `' ( R ^ r i )  =  `' ( R ^ r ( n  +  1 ) ) )
19 oveq2 5882 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  ( `' R ^ r i )  =  ( `' R ^ r ( n  +  1 ) ) )
2018, 19eqeq12d 2310 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) ) )
2116, 20imbi12d 311 . . . 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 2356 . . . . . 6  |-  ( i  =  N  ->  (
i  e.  NN0  <->  N  e.  NN0 ) )
2322anbi1d 685 . . . . 5  |-  ( i  =  N  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( N  e.  NN0  /\  ph )
) )
24 oveq2 5882 . . . . . . 7  |-  ( i  =  N  ->  ( R ^ r i )  =  ( R ^
r N ) )
2524cnveqd 4873 . . . . . 6  |-  ( i  =  N  ->  `' ( R ^ r i )  =  `' ( R ^ r N ) )
26 oveq2 5882 . . . . . 6  |-  ( i  =  N  ->  ( `' R ^ r i )  =  ( `' R ^ r N ) )
2725, 26eqeq12d 2310 . . . . 5  |-  ( i  =  N  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
2823, 27imbi12d 311 . . . 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 24040 . . . . . 6  |-  ( ph  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R
) )
3231adantl 452 . . . . 5  |-  ( ( 0  e.  NN0  /\  ph )  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R ) )
33 id 19 . . . . . . 7  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  ( R ^
r 0 )  =  (  _I  |`  U. U. R ) )
3433cnveqd 4873 . . . . . 6  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  `' ( R ^ r 0 )  =  `' (  _I  |`  U. U. R ) )
35 cnvresid 5338 . . . . . . 7  |-  `' (  _I  |`  U. U. R
)  =  (  _I  |`  U. U. R )
3629adantl 452 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  Rel  R )
37 relcnvfld 5219 . . . . . . . . 9  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
3836, 37syl 15 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  ph )  ->  U. U. R  =  U. U. `' R
)
39 reseq2 4966 . . . . . . . . 9  |-  ( U. U. R  =  U. U. `' R  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
40 simpr 447 . . . . . . . . . . 11  |-  ( ( 0  e.  NN0  /\  ph )  ->  ph )
41 relcnv 5067 . . . . . . . . . . 11  |-  Rel  `' R
42 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  Rel  `' R
)
43 simpl 443 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Rel  `' R
)  ->  ph )
44 cnvexg 5224 . . . . . . . . . . . . 13  |-  ( R  e.  _V  ->  `' R  e.  _V )
4543, 30, 443syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  `' R  e.  _V )
4642, 45relexp0 24040 . . . . . . . . . . 11  |-  ( (
ph  /\  Rel  `' R
)  ->  ( `' R ^ r 0 )  =  (  _I  |`  U. U. `' R ) )
4740, 41, 46sylancl 643 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  ph )  ->  ( `' R ^ r 0 )  =  (  _I  |`  U. U. `' R ) )
4847eqcomd 2301 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. `' R )  =  ( `' R ^ r 0 ) )
4939, 48sylan9eq 2348 . . . . . . . 8  |-  ( ( U. U. R  = 
U. U. `' R  /\  ( 0  e.  NN0  /\ 
ph ) )  -> 
(  _I  |`  U. U. R )  =  ( `' R ^ r 0 ) )
5038, 49mpancom 650 . . . . . . 7  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. R )  =  ( `' R ^
r 0 ) )
5135, 50syl5eq 2340 . . . . . 6  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' (  _I  |`  U. U. R
)  =  ( `' R ^ r 0 ) )
5234, 51sylan9eq 2348 . . . . 5  |-  ( ( ( R ^ r 0 )  =  (  _I  |`  U. U. R
)  /\  ( 0  e.  NN0  /\  ph )
)  ->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) )
5332, 52mpancom 650 . . . 4  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) )
54 simprrr 741 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  n  e.  NN0 )
55 simpl 443 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) )  ->  ph )
5655adantl 452 . . . . . . . . . . . 12  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ph )
5756, 29syl 15 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  Rel  R )
5856, 30syl 15 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  R  e.  _V )
5957, 58relexpsucl 24043 . . . . . . . . . 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 14 . . . . . . . . 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 4873 . . . . . . . 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 19 . . . . . . . . . 10  |-  ( `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^ r n ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^
r n ) ) )
63 cnvco 4881 . . . . . . . . . . 11  |-  `' ( R  o.  ( R ^ r n ) )  =  ( `' ( R ^ r n )  o.  `' R )
64 simprrl 740 . . . . . . . . . . . . 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 660 . . . . . . . . . . . 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 4861 . . . . . . . . . . 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 2340 . . . . . . . . . 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 2348 . . . . . . . . 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 452 . . . . . . . . . 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 10 . . . . . . . . . . 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 740 . . . . . . . . . . . 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 18 . . . . . . . . . . 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 24041 . . . . . . . . . 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 14 . . . . . . . . 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 2331 . . . . . . . 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 650 . . . . . . 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 629 . . . . . 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 424 . . . . 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 424 . . . 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 10124 . . 3  |-  ( N  e.  NN0  ->  ( ( N  e.  NN0  /\  ph )  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
8180anabsi5 790 . 2  |-  ( ( N  e.  NN0  /\  ph )  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) )
8281expcom 424 1  |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   U.cuni 3843    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   Rel wrel 4710  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   NN0cn0 9981   ^ rcrelexp 24038
This theorem is referenced by:  relexprel  24046
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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-nel 2462  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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-relexp 24039
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