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Theorem relexp01min 36376
Description: With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
Assertion
Ref Expression
relexp01min  |-  ( ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  /\  ( J  e.  { 0 ,  1 }  /\  K  e. 
{ 0 ,  1 } ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )

Proof of Theorem relexp01min
StepHypRef Expression
1 elpri 3976 . . 3  |-  ( J  e.  { 0 ,  1 }  ->  ( J  =  0  \/  J  =  1 ) )
2 elpri 3976 . . 3  |-  ( K  e.  { 0 ,  1 }  ->  ( K  =  0  \/  K  =  1 ) )
3 dmresi 5166 . . . . . . . . . . 11  |-  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
4 rnresi 5187 . . . . . . . . . . 11  |-  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
53, 4uneq12i 3577 . . . . . . . . . 10  |-  ( dom  (  _I  |`  ( dom  R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) )  =  ( ( dom  R  u.  ran  R )  u.  ( dom  R  u.  ran  R ) )
6 unidm 3568 . . . . . . . . . 10  |-  ( ( dom  R  u.  ran  R )  u.  ( dom 
R  u.  ran  R
) )  =  ( dom  R  u.  ran  R )
75, 6eqtri 2493 . . . . . . . . 9  |-  ( dom  (  _I  |`  ( dom  R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) )  =  ( dom  R  u.  ran  R )
87reseq2i 5108 . . . . . . . 8  |-  (  _I  |`  ( dom  (  _I  |`  ( dom  R  u.  ran  R ) )  u. 
ran  (  _I  |`  ( dom  R  u.  ran  R
) ) ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) )
9 simp1 1030 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  0 )
109oveq2d 6324 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  0 ) )
11 simp3l 1058 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  R  e.  V )
12 relexp0g 13162 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
1311, 12syl 17 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
1410, 13eqtrd 2505 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
15 simp2 1031 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  0 )
1614, 15oveq12d 6326 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( (  _I  |`  ( dom  R  u.  ran  R
) ) ^r 
0 ) )
17 dmexg 6743 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  dom  R  e.  _V )
18 rnexg 6744 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  ran  R  e.  _V )
19 unexg 6611 . . . . . . . . . . . 12  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  ( dom  R  u.  ran  R )  e. 
_V )
2017, 18, 19syl2anc 673 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( dom  R  u.  ran  R
)  e.  _V )
2120resiexd 6147 . . . . . . . . . 10  |-  ( R  e.  V  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  e.  _V )
22 relexp0g 13162 . . . . . . . . . 10  |-  ( (  _I  |`  ( dom  R  u.  ran  R ) )  e.  _V  ->  ( (  _I  |`  ( dom  R  u.  ran  R
) ) ^r 
0 )  =  (  _I  |`  ( dom  (  _I  |`  ( dom 
R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) ) ) )
2311, 21, 223syl 18 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
(  _I  |`  ( dom  R  u.  ran  R
) ) ^r 
0 )  =  (  _I  |`  ( dom  (  _I  |`  ( dom 
R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) ) ) )
2416, 23eqtrd 2505 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  (  _I  |`  ( dom  (  _I  |`  ( dom 
R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) ) ) )
25 simp3r 1059 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
26 0re 9661 . . . . . . . . . . . . . 14  |-  0  e.  RR
2726ltnri 9761 . . . . . . . . . . . . 13  |-  -.  0  <  0
289, 15breq12d 4408 . . . . . . . . . . . . 13  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  0  <  0 ) )
2927, 28mtbiri 310 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  J  <  K )
3029iffalsed 3883 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  K )
3125, 30, 153eqtrd 2509 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  0 )
3231oveq2d 6324 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  0 ) )
3332, 13eqtrd 2505 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
348, 24, 333eqtr4a 2531 . . . . . . 7  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
35343exp 1230 . . . . . 6  |-  ( J  =  0  ->  ( K  =  0  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
36 simp1 1030 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  1 )
3736oveq2d 6324 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  1 ) )
38 simp3l 1058 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  R  e.  V )
39 relexp1g 13166 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( R ^r  1 )  =  R )
4038, 39syl 17 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  1 )  =  R )
4137, 40eqtrd 2505 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  R )
42 simp2 1031 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  0 )
4341, 42oveq12d 6326 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  0 ) )
44 simp3r 1059 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
45 0lt1 10157 . . . . . . . . . . . . 13  |-  0  <  1
46 1re 9660 . . . . . . . . . . . . . 14  |-  1  e.  RR
4726, 46ltnsymi 9771 . . . . . . . . . . . . 13  |-  ( 0  <  1  ->  -.  1  <  0 )
4845, 47mp1i 13 . . . . . . . . . . . 12  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  1  <  0 )
4936, 42breq12d 4408 . . . . . . . . . . . 12  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  1  <  0 ) )
5048, 49mtbird 308 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  J  <  K )
5150iffalsed 3883 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  K )
5244, 51, 423eqtrd 2509 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  0 )
5352oveq2d 6324 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  0 ) )
5443, 53eqtr4d 2508 . . . . . . 7  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
55543exp 1230 . . . . . 6  |-  ( J  =  1  ->  ( K  =  0  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
5635, 55jaoi 386 . . . . 5  |-  ( ( J  =  0  \/  J  =  1 )  ->  ( K  =  0  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
57 ovex 6336 . . . . . . . . 9  |-  ( R ^r  0 )  e.  _V
58 relexp1g 13166 . . . . . . . . 9  |-  ( ( R ^r  0 )  e.  _V  ->  ( ( R ^r 
0 ) ^r 
1 )  =  ( R ^r  0 ) )
5957, 58mp1i 13 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r 
0 ) ^r 
1 )  =  ( R ^r  0 ) )
60 simp1 1030 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  0 )
6160oveq2d 6324 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  0 ) )
62 simp2 1031 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  1 )
6361, 62oveq12d 6326 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( ( R ^r 
0 ) ^r 
1 ) )
64 simp3r 1059 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
6560, 62breq12d 4408 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  0  <  1 ) )
6645, 65mpbiri 241 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  <  K )
6766iftrued 3880 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  J )
6864, 67, 603eqtrd 2509 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  0 )
6968oveq2d 6324 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  0 ) )
7059, 63, 693eqtr4d 2515 . . . . . . 7  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
71703exp 1230 . . . . . 6  |-  ( J  =  0  ->  ( K  =  1  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
72 simp1 1030 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  1 )
7372oveq2d 6324 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  1 ) )
74 simp3l 1058 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  R  e.  V )
7574, 39syl 17 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  1 )  =  R )
7673, 75eqtrd 2505 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  R )
77 simp2 1031 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  1 )
7876, 77oveq12d 6326 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  1 ) )
79 simp3r 1059 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
8046ltnri 9761 . . . . . . . . . . . 12  |-  -.  1  <  1
8172, 77breq12d 4408 . . . . . . . . . . . 12  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  1  <  1 ) )
8280, 81mtbiri 310 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  J  <  K )
8382iffalsed 3883 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  K )
8479, 83, 773eqtrd 2509 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  1 )
8584oveq2d 6324 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  1 ) )
8678, 85eqtr4d 2508 . . . . . . 7  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
87863exp 1230 . . . . . 6  |-  ( J  =  1  ->  ( K  =  1  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
8871, 87jaoi 386 . . . . 5  |-  ( ( J  =  0  \/  J  =  1 )  ->  ( K  =  1  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
8956, 88jaod 387 . . . 4  |-  ( ( J  =  0  \/  J  =  1 )  ->  ( ( K  =  0  \/  K  =  1 )  -> 
( ( R  e.  V  /\  I  =  if ( J  < 
K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
9089imp 436 . . 3  |-  ( ( ( J  =  0  \/  J  =  1 )  /\  ( K  =  0  \/  K  =  1 ) )  ->  ( ( R  e.  V  /\  I  =  if ( J  < 
K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) )
911, 2, 90syl2an 485 . 2  |-  ( ( J  e.  { 0 ,  1 }  /\  K  e.  { 0 ,  1 } )  ->  ( ( R  e.  V  /\  I  =  if ( J  < 
K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) )
9291impcom 437 1  |-  ( ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  /\  ( J  e.  { 0 ,  1 }  /\  K  e. 
{ 0 ,  1 } ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388   ifcif 3872   {cpr 3961   class class class wbr 4395    _I cid 4749   dom cdm 4839   ran crn 4840    |` cres 4841  (class class class)co 6308   0cc0 9557   1c1 9558    < clt 9693   ^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-rep 4508  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:  relexp1idm  36377  relexp0idm  36378
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