Theorem relexpiidm 37996

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.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 |- ( x = 0 -> ( ( _I |` A ) ^r x ) = ( ( _I |` A ) ^r 0 ) )
2	1	eqeq1d 2624	. . . 4 |- ( x = 0 -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) ) )
3	2	imbi2d 330	. . 3 |- ( x = 0 -> ( ( A e. V -> ( ( _I |` A ) ^r x ) = ( _I |` A ) ) <-> ( A e. V -> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) ) ) )
4		oveq2 6658	. . . . 5 |- ( x = y -> ( ( _I |` A ) ^r x ) = ( ( _I |` A ) ^r y ) )
5	4	eqeq1d 2624	. . . 4 |- ( x = y -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r y ) = ( _I |` A ) ) )
6	5	imbi2d 330	. . 3 |- ( x = y -> ( ( A e. V -> ( ( _I |` A ) ^r x ) = ( _I |` A ) ) <-> ( A e. V -> ( ( _I |` A ) ^r y ) = ( _I |` A ) ) ) )
7		oveq2 6658	. . . . 5 |- ( x = ( y + 1 ) -> ( ( _I |` A ) ^r x ) = ( ( _I |` A ) ^r ( y + 1 ) ) )
8	7	eqeq1d 2624	. . . 4 |- ( x = ( y + 1 ) -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) )
9	8	imbi2d 330	. . 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 6658	. . . . 5 |- ( x = N -> ( ( _I |` A ) ^r x ) = ( ( _I |` A ) ^r N ) )
11	10	eqeq1d 2624	. . . 4 |- ( x = N -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) )
12	11	imbi2d 330	. . 3 |- ( x = N -> ( ( A e. V -> ( ( _I |` A ) ^r x ) = ( _I |` A ) ) <-> ( A e. V -> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) ) )
13		resiexg 7102	. . . . 5 |- ( A e. V -> ( _I |` A ) e. _V )
14		relexp0g 13762	. . . . 5 |- ( ( _I |` A ) e. _V -> ( ( _I |` A ) ^r 0 ) = ( _I |` ( dom ( _I |` A ) u. ran ( _I |` A ) ) ) )
15	13, 14	syl 17	. . . 4 |- ( A e. V -> ( ( _I |` A ) ^r 0 ) = ( _I |` ( dom ( _I |` A ) u. ran ( _I |` A ) ) ) )
16		dmresi 5457	. . . . . . 7 |- dom ( _I |` A ) = A
17		rnresi 5479	. . . . . . 7 |- ran ( _I |` A ) = A
18	16, 17	uneq12i 3765	. . . . . 6 |- ( dom ( _I |` A ) u. ran ( _I |` A ) ) = ( A u. A )
19		unidm 3756	. . . . . 6 |- ( A u. A ) = A
20	18, 19	eqtri 2644	. . . . 5 |- ( dom ( _I |` A ) u. ran ( _I |` A ) ) = A
21	20	reseq2i 5393	. . . 4 |- ( _I |` ( dom ( _I |` A ) u. ran ( _I |` A ) ) ) = ( _I |` A )
22	15, 21	syl6eq 2672	. . 3 |- ( A e. V -> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) )
23		relres 5426	. . . . . . . . . 10 |- Rel ( _I |` A )
24	23	a1i 11	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V ) -> Rel ( _I |` A ) )
25	13	adantl 482	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V ) -> ( _I |` A ) e. _V )
26	24, 25	relexpsucrd 13770	. . . . . . . 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 ) ) ) )
27	26	3impia 1261	. . . . . . 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 1061	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V /\ y e. NN0 ) -> ( ( _I |` A ) ^r y ) = ( _I |` A ) )
29	28	coeq1d 5283	. . . . . . . 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 5653	. . . . . . . . 9 |- ( ( _I |` A ) o. ( _I |` A ) ) = ( ( _I |` A ) |` A )
31		residm 5430	. . . . . . . . 9 |- ( ( _I |` A ) |` A ) = ( _I |` A )
32	30, 31	eqtri 2644	. . . . . . . 8 |- ( ( _I |` A ) o. ( _I |` A ) ) = ( _I |` A )
33	29, 32	syl6eq 2672	. . . . . . 7 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V /\ y e. NN0 ) -> ( ( ( _I |` A ) ^r y ) o. ( _I |` A ) ) = ( _I |` A ) )
34	27, 33	eqtrd 2656	. . . . . 6 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V /\ y e. NN0 ) -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) )
35	34	3exp 1264	. . . . 5 |- ( ( ( _I |` A ) ^r y ) = ( _I |` A ) -> ( A e. V -> ( y e. NN0 -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) ) )
36	35	com13 88	. . . 4 |- ( y e. NN0 -> ( A e. V -> ( ( ( _I |` A ) ^r y ) = ( _I |` A ) -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) ) )
37	36	a2d 29	. . 3 |- ( y e. NN0 -> ( ( A e. V -> ( ( _I |` A ) ^r y ) = ( _I |` A ) ) -> ( A e. V -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) ) )
38	3, 6, 9, 12, 22, 37	nn0ind 11472	. 2 |- ( N e. NN0 -> ( A e. V -> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) )
39	38	impcom 446	1 |- ( ( A e. V /\ N e. NN0 ) -> ( ( _I |` A ) ^r N ) = ( _I |` A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ^r crelexp 13760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-relexp 13761

This theorem is referenced by:  relexpmulg  38002  relexpxpmin  38009