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Theorem relexpiidm 36296
Description: Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
Assertion
Ref Expression
relexpiidm  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) )

Proof of Theorem relexpiidm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6298 . . . . 5  |-  ( x  =  0  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r 
0 ) )
21eqeq1d 2453 . . . 4  |-  ( x  =  0  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  0 )  =  (  _I  |`  A ) ) )
32imbi2d 318 . . 3  |-  ( x  =  0  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  0 )  =  (  _I  |`  A ) ) ) )
4 oveq2 6298 . . . . 5  |-  ( x  =  y  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r 
y ) )
54eqeq1d 2453 . . . 4  |-  ( x  =  y  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A ) ) )
65imbi2d 318 . . 3  |-  ( x  =  y  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A ) ) ) )
7 oveq2 6298 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r 
( y  +  1 ) ) )
87eqeq1d 2453 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  (  _I  |`  A ) ) )
98imbi2d 318 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  (  _I  |`  A ) ) ) )
10 oveq2 6298 . . . . 5  |-  ( x  =  N  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r  N ) )
1110eqeq1d 2453 . . . 4  |-  ( x  =  N  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) ) )
1211imbi2d 318 . . 3  |-  ( x  =  N  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A )
) ) )
13 resiexg 6729 . . . . 5  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
14 relexp0g 13085 . . . . 5  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A ) ^r  0 )  =  (  _I  |`  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) ) ) )
1513, 14syl 17 . . . 4  |-  ( A  e.  V  ->  (
(  _I  |`  A ) ^r  0 )  =  (  _I  |`  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) ) ) )
16 dmresi 5160 . . . . . . 7  |-  dom  (  _I  |`  A )  =  A
17 rnresi 5181 . . . . . . 7  |-  ran  (  _I  |`  A )  =  A
1816, 17uneq12i 3586 . . . . . 6  |-  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) )  =  ( A  u.  A
)
19 unidm 3577 . . . . . 6  |-  ( A  u.  A )  =  A
2018, 19eqtri 2473 . . . . 5  |-  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) )  =  A
2120reseq2i 5102 . . . 4  |-  (  _I  |`  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) ) )  =  (  _I  |`  A )
2215, 21syl6eq 2501 . . 3  |-  ( A  e.  V  ->  (
(  _I  |`  A ) ^r  0 )  =  (  _I  |`  A ) )
23 relres 5132 . . . . . . . . . 10  |-  Rel  (  _I  |`  A )
2423a1i 11 . . . . . . . . 9  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V
)  ->  Rel  (  _I  |`  A ) )
2513adantl 468 . . . . . . . . 9  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V
)  ->  (  _I  |`  A )  e.  _V )
2624, 25relexpsucrd 13093 . . . . . . . 8  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V
)  ->  ( y  e.  NN0  ->  ( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  ( ( (  _I  |`  A ) ^r 
y )  o.  (  _I  |`  A ) ) ) )
27263impia 1205 . . . . . . 7  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( (  _I  |`  A ) ^r 
( y  +  1 ) )  =  ( ( (  _I  |`  A ) ^r  y )  o.  (  _I  |`  A ) ) )
28 simp1 1008 . . . . . . . . 9  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( (  _I  |`  A ) ^r 
y )  =  (  _I  |`  A )
)
2928coeq1d 4996 . . . . . . . 8  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( ( (  _I  |`  A ) ^r  y )  o.  (  _I  |`  A ) )  =  ( (  _I  |`  A )  o.  (  _I  |`  A ) ) )
30 coires1 5353 . . . . . . . . 9  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  =  ( (  _I  |`  A )  |`  A )
31 residm 5136 . . . . . . . . 9  |-  ( (  _I  |`  A )  |`  A )  =  (  _I  |`  A )
3230, 31eqtri 2473 . . . . . . . 8  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  =  (  _I  |`  A )
3329, 32syl6eq 2501 . . . . . . 7  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( ( (  _I  |`  A ) ^r  y )  o.  (  _I  |`  A ) )  =  (  _I  |`  A ) )
3427, 33eqtrd 2485 . . . . . 6  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( (  _I  |`  A ) ^r 
( y  +  1 ) )  =  (  _I  |`  A )
)
35343exp 1207 . . . . 5  |-  ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  ->  ( A  e.  V  ->  ( y  e.  NN0  ->  ( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  (  _I  |`  A ) ) ) )
3635com13 83 . . . 4  |-  ( y  e.  NN0  ->  ( A  e.  V  ->  (
( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  ->  ( (  _I  |`  A ) ^r 
( y  +  1 ) )  =  (  _I  |`  A )
) ) )
3736a2d 29 . . 3  |-  ( y  e.  NN0  ->  ( ( A  e.  V  -> 
( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A ) )  ->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  (  _I  |`  A ) ) ) )
383, 6, 9, 12, 22, 37nn0ind 11030 . 2  |-  ( N  e.  NN0  ->  ( A  e.  V  ->  (
(  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) ) )
3938impcom 432 1  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045    u. cun 3402    _I cid 4744   dom cdm 4834   ran crn 4835    |` cres 4836    o. ccom 4838   Rel wrel 4839  (class class class)co 6290   0cc0 9539   1c1 9540    + caddc 9542   NN0cn0 10869   ^r crelexp 13083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-seq 12214  df-relexp 13084
This theorem is referenced by:  relexpmulg  36302  relexpxpmin  36309
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