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.

Step	Hyp	Ref	Expression
1		oveq2 6316	. . . . 5 |- ( x = 0 -> ( ( _I |` A ) ^r x ) = ( ( _I |` A ) ^r 0 ) )
2	1	eqeq1d 2473	. . . 4 |- ( x = 0 -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) ) )
3	2	imbi2d 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 ) )
5	4	eqeq1d 2473	. . . 4 |- ( x = y -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r y ) = ( _I |` A ) ) )
6	5	imbi2d 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 ) ) )
8	7	eqeq1d 2473	. . . 4 |- ( x = ( y + 1 ) -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) )
9	8	imbi2d 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 ) )
11	10	eqeq1d 2473	. . . 4 |- ( x = N -> ( ( ( _I |` A ) ^r x ) = ( _I |` A ) <-> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) )
12	11	imbi2d 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 ) ) ) )
15	13, 14	syl 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
18	16, 17	uneq12i 3577	. . . . . 6 |- ( dom ( _I |` A ) u. ran ( _I |` A ) ) = ( A u. A )
19		unidm 3568	. . . . . 6 |- ( A u. A ) = A
20	18, 19	eqtri 2493	. . . . 5 |- ( dom ( _I |` A ) u. ran ( _I |` A ) ) = A
21	20	reseq2i 5108	. . . 4 |- ( _I |` ( dom ( _I |` A ) u. ran ( _I |` A ) ) ) = ( _I |` A )
22	15, 21	syl6eq 2521	. . 3 |- ( A e. V -> ( ( _I |` A ) ^r 0 ) = ( _I |` A ) )
23		relres 5138	. . . . . . . . . 10 |- Rel ( _I |` A )
24	23	a1i 11	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V ) -> Rel ( _I |` A ) )
25	13	adantl 473	. . . . . . . . 9 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V ) -> ( _I |` A ) e. _V )
26	24, 25	relexpsucrd 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 ) ) ) )
27	26	3impia 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 ) )
29	28	coeq1d 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 )
32	30, 31	eqtri 2493	. . . . . . . 8 |- ( ( _I |` A ) o. ( _I |` A ) ) = ( _I |` A )
33	29, 32	syl6eq 2521	. . . . . . 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 2505	. . . . . 6 |- ( ( ( ( _I |` A ) ^r y ) = ( _I |` A ) /\ A e. V /\ y e. NN0 ) -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) )
35	34	3exp 1230	. . . . 5 |- ( ( ( _I |` A ) ^r y ) = ( _I |` A ) -> ( A e. V -> ( y e. NN0 -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) ) )
36	35	com13 82	. . . 4 |- ( y e. NN0 -> ( A e. V -> ( ( ( _I |` A ) ^r y ) = ( _I |` A ) -> ( ( _I |` A ) ^r ( y + 1 ) ) = ( _I |` A ) ) ) )
37	36	a2d 28	. . 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 11053	. 2 |- ( N e. NN0 -> ( A e. V -> ( ( _I |` A ) ^r N ) = ( _I |` A ) ) )
39	38	impcom 437	1 |- ( ( A e. V /\ N e. NN0 ) -> ( ( _I |` A ) ^r N ) = ( _I |` A ) )

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