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.

Step	Hyp	Ref	Expression
1		oveq2 6298	. . . . 5 |- ( x = 0 -> ( ( _I |` A ) ^r x ) = ( ( _I |` A ) ^r 0 ) )
2	1	eqeq1d 2453	. . . 4 |- ( x = 0 -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) ) )
3	2	imbi2d 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 ) )
5	4	eqeq1d 2453	. . . 4 |- ( x = y -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r y ) = ( _I |` A ) ) )
6	5	imbi2d 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 ) ) )
8	7	eqeq1d 2453	. . . 4 |- ( x = ( y + 1 ) -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) )
9	8	imbi2d 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 ) )
11	10	eqeq1d 2453	. . . 4 |- ( x = N -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) )
12	11	imbi2d 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 ) ) ) )
15	13, 14	syl 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
18	16, 17	uneq12i 3586	. . . . . 6 |- ( dom ( _I |` A ) u. ran ( _I |` A ) ) = ( A u. A )
19		unidm 3577	. . . . . 6 |- ( A u. A ) = A
20	18, 19	eqtri 2473	. . . . 5 |- ( dom ( _I |` A ) u. ran ( _I |` A ) ) = A
21	20	reseq2i 5102	. . . 4 |- ( _I |` ( dom ( _I |` A ) u. ran ( _I |` A ) ) ) = ( _I |` A )
22	15, 21	syl6eq 2501	. . 3 |- ( A e. V -> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) )
23		relres 5132	. . . . . . . . . 10 |- Rel ( _I |` A )
24	23	a1i 11	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V ) -> Rel ( _I |` A ) )
25	13	adantl 468	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V ) -> ( _I |` A ) e. _V )
26	24, 25	relexpsucrd 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 ) ) ) )
27	26	3impia 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 ) )
29	28	coeq1d 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 )
32	30, 31	eqtri 2473	. . . . . . . 8 |- ( ( _I |` A ) o. ( _I |` A ) ) = ( _I |` A )
33	29, 32	syl6eq 2501	. . . . . . 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 2485	. . . . . 6 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V /\ y e. NN0 ) -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) )
35	34	3exp 1207	. . . . 5 |- ( ( ( _I |` A ) ^r y ) = ( _I |` A ) -> ( A e. V -> ( y e. NN0 -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) ) )
36	35	com13 83	. . . 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 11030	. 2 |- ( N e. NN0 -> ( A e. V -> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) )
39	38	impcom 432	1 |- ( ( A e. V /\ N e. NN0 ) -> ( ( _I |` A ) ^r N ) = ( _I |` A ) )

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