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Theorem relexp0g 12939
Description: A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
Assertion
Ref Expression
relexp0g  |-  ( R  e.  V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )

Proof of Theorem relexp0g
Dummy variables  n  r  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2455 . . 3  |-  ( R  e.  V  ->  (
r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) ) )  =  ( r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) ) ) )
2 simprr 755 . . . . 5  |-  ( ( R  e.  V  /\  ( r  =  R  /\  n  =  0 ) )  ->  n  =  0 )
32iftrued 3937 . . . 4  |-  ( ( R  e.  V  /\  ( r  =  R  /\  n  =  0 ) )  ->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) )  =  (  _I  |`  ( dom  r  u.  ran  r ) ) )
4 dmeq 5192 . . . . . . 7  |-  ( r  =  R  ->  dom  r  =  dom  R )
5 rneq 5217 . . . . . . 7  |-  ( r  =  R  ->  ran  r  =  ran  R )
64, 5uneq12d 3645 . . . . . 6  |-  ( r  =  R  ->  ( dom  r  u.  ran  r )  =  ( dom  R  u.  ran  R ) )
76reseq2d 5262 . . . . 5  |-  ( r  =  R  ->  (  _I  |`  ( dom  r  u.  ran  r ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
87ad2antrl 725 . . . 4  |-  ( ( R  e.  V  /\  ( r  =  R  /\  n  =  0 ) )  ->  (  _I  |`  ( dom  r  u.  ran  r ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
93, 8eqtrd 2495 . . 3  |-  ( ( R  e.  V  /\  ( r  =  R  /\  n  =  0 ) )  ->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) )  =  (  _I  |`  ( dom  R  u.  ran  R ) ) )
10 elex 3115 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
11 0nn0 10806 . . . 4  |-  0  e.  NN0
1211a1i 11 . . 3  |-  ( R  e.  V  ->  0  e.  NN0 )
13 dmexg 6704 . . . . 5  |-  ( R  e.  V  ->  dom  R  e.  _V )
14 rnexg 6705 . . . . 5  |-  ( R  e.  V  ->  ran  R  e.  _V )
15 unexg 6574 . . . . 5  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  ( dom  R  u.  ran  R )  e. 
_V )
1613, 14, 15syl2anc 659 . . . 4  |-  ( R  e.  V  ->  ( dom  R  u.  ran  R
)  e.  _V )
17 resiexg 6709 . . . 4  |-  ( ( dom  R  u.  ran  R )  e.  _V  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  e.  _V )
1816, 17syl 16 . . 3  |-  ( R  e.  V  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  e.  _V )
191, 9, 10, 12, 18ovmpt2d 6403 . 2  |-  ( R  e.  V  ->  ( R ( r  e. 
_V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) ) 0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
20 df-relexp 12938 . . 3  |- ^r 
=  ( r  e. 
_V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )
21 oveq 6276 . . . . 5  |-  ( ^r  =  ( r  e.  _V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )  ->  ( R ^r  0 )  =  ( R ( r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) ) ) 0 ) )
2221eqeq1d 2456 . . . 4  |-  ( ^r  =  ( r  e.  _V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )  ->  ( ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) )  <->  ( R
( r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) ) ) 0 )  =  (  _I  |`  ( dom  R  u.  ran  R ) ) ) )
2322imbi2d 314 . . 3  |-  ( ^r  =  ( r  e.  _V ,  n  e. 
NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) )  ->  ( ( R  e.  V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R ) ) )  <-> 
( R  e.  V  ->  ( R ( r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e. 
_V  |->  r ) ) `
 n ) ) ) 0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) ) ) )
2420, 23ax-mp 5 . 2  |-  ( ( R  e.  V  -> 
( R ^r 
0 )  =  (  _I  |`  ( dom  R  u.  ran  R ) ) )  <->  ( R  e.  V  ->  ( R ( r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r ) ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r
) ) ,  ( z  e.  _V  |->  r ) ) `  n
) ) ) 0 )  =  (  _I  |`  ( dom  R  u.  ran  R ) ) ) )
2519, 24mpbir 209 1  |-  ( R  e.  V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459   ifcif 3929    |-> cmpt 4497    _I cid 4779   dom cdm 4988   ran crn 4989    |` cres 4990    o. ccom 4992   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481   1c1 9482   NN0cn0 10791    seqcseq 12089   ^r crelexp 12937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-mulcl 9543  ax-i2m1 9549
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-n0 10792  df-relexp 12938
This theorem is referenced by:  relexp0  12940  relexpcnv  12950  relexp0rel  12952  relexpdmg  12957  relexprng  12961  relexpfld  12964  relexpaddg  12968  dfrcl3  38194  relexpaddss  38205  iunrelexpmin2  38208  dfrtrcl3  38214  relexpss1d  38217  relexp0eq  38218  relexp01min  38219  relexpiidm  38222  relexp0a  38223  iunrelexp0  38224  relexpxpmin  38226  relexpmulg  38228  cotrclrcl  38232
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