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Theorem relexpiidm 36367
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 6316 . . . . 5  |-  ( x  =  0  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r 
0 ) )
21eqeq1d 2473 . . . 4  |-  ( x  =  0  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  0 )  =  (  _I  |`  A ) ) )
32imbi2d 323 . . 3  |-  ( x  =  0  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  0 )  =  (  _I  |`  A ) ) ) )
4 oveq2 6316 . . . . 5  |-  ( x  =  y  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r 
y ) )
54eqeq1d 2473 . . . 4  |-  ( x  =  y  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A ) ) )
65imbi2d 323 . . 3  |-  ( x  =  y  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A ) ) ) )
7 oveq2 6316 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r 
( y  +  1 ) ) )
87eqeq1d 2473 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  (  _I  |`  A ) ) )
98imbi2d 323 . . 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 6316 . . . . 5  |-  ( x  =  N  ->  (
(  _I  |`  A ) ^r  x )  =  ( (  _I  |`  A ) ^r  N ) )
1110eqeq1d 2473 . . . 4  |-  ( x  =  N  ->  (
( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A )  <-> 
( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) ) )
1211imbi2d 323 . . 3  |-  ( x  =  N  ->  (
( A  e.  V  ->  ( (  _I  |`  A ) ^r  x )  =  (  _I  |`  A ) )  <->  ( A  e.  V  ->  ( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A )
) ) )
13 resiexg 6748 . . . . 5  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
14 relexp0g 13162 . . . . 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 5166 . . . . . . 7  |-  dom  (  _I  |`  A )  =  A
17 rnresi 5187 . . . . . . 7  |-  ran  (  _I  |`  A )  =  A
1816, 17uneq12i 3577 . . . . . 6  |-  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) )  =  ( A  u.  A
)
19 unidm 3568 . . . . . 6  |-  ( A  u.  A )  =  A
2018, 19eqtri 2493 . . . . 5  |-  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) )  =  A
2120reseq2i 5108 . . . 4  |-  (  _I  |`  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) ) )  =  (  _I  |`  A )
2215, 21syl6eq 2521 . . 3  |-  ( A  e.  V  ->  (
(  _I  |`  A ) ^r  0 )  =  (  _I  |`  A ) )
23 relres 5138 . . . . . . . . . 10  |-  Rel  (  _I  |`  A )
2423a1i 11 . . . . . . . . 9  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V
)  ->  Rel  (  _I  |`  A ) )
2513adantl 473 . . . . . . . . 9  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V
)  ->  (  _I  |`  A )  e.  _V )
2624, 25relexpsucrd 13170 . . . . . . . 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 1228 . . . . . . 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 1030 . . . . . . . . 9  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( (  _I  |`  A ) ^r 
y )  =  (  _I  |`  A )
)
2928coeq1d 5001 . . . . . . . 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 5360 . . . . . . . . 9  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  =  ( (  _I  |`  A )  |`  A )
31 residm 5142 . . . . . . . . 9  |-  ( (  _I  |`  A )  |`  A )  =  (  _I  |`  A )
3230, 31eqtri 2493 . . . . . . . 8  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  =  (  _I  |`  A )
3329, 32syl6eq 2521 . . . . . . 7  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( ( (  _I  |`  A ) ^r  y )  o.  (  _I  |`  A ) )  =  (  _I  |`  A ) )
3427, 33eqtrd 2505 . . . . . 6  |-  ( ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  /\  A  e.  V  /\  y  e.  NN0 )  ->  ( (  _I  |`  A ) ^r 
( y  +  1 ) )  =  (  _I  |`  A )
)
35343exp 1230 . . . . 5  |-  ( ( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  ->  ( A  e.  V  ->  ( y  e.  NN0  ->  ( (  _I  |`  A ) ^r  ( y  +  1 ) )  =  (  _I  |`  A ) ) ) )
3635com13 82 . . . 4  |-  ( y  e.  NN0  ->  ( A  e.  V  ->  (
( (  _I  |`  A ) ^r  y )  =  (  _I  |`  A )  ->  ( (  _I  |`  A ) ^r 
( y  +  1 ) )  =  (  _I  |`  A )
) ) )
3736a2d 28 . . 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 11053 . 2  |-  ( N  e.  NN0  ->  ( A  e.  V  ->  (
(  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) ) )
3938impcom 437 1  |-  ( ( A  e.  V  /\  N  e.  NN0 )  -> 
( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388    _I cid 4749   dom cdm 4839   ran crn 4840    |` cres 4841    o. ccom 4843   Rel wrel 4844  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560   NN0cn0 10893   ^r crelexp 13160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-seq 12252  df-relexp 13161
This theorem is referenced by:  relexpmulg  36373  relexpxpmin  36380
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