Users' Mathboxes Mathbox for Drahflow < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfrtrclrec2 Unicode version

Theorem dfrtrclrec2 24055
Description: If two elements are connected by a reflexive, transitive closure, then they are connected via  n instances the relation, for some  n. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
rtrclreclem.1  |-  ( ph  ->  Rel  R )
rtrclreclem.2  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
dfrtrclrec2  |-  ( ph  ->  ( A ( t *rec `  R ) B 
<->  E. n  e.  NN0  A ( R ^ r n ) B ) )
Distinct variable groups:    R, n    A, n    B, n
Allowed substitution hint:    ph( n)

Proof of Theorem dfrtrclrec2
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 rtrclreclem.2 . . . 4  |-  ( ph  ->  R  e.  _V )
2 nn0ex 9987 . . . . 5  |-  NN0  e.  _V
3 ovex 5899 . . . . 5  |-  ( R ^ r n )  e.  _V
42, 3iunex 5786 . . . 4  |-  U_ n  e.  NN0  ( R ^
r n )  e. 
_V
5 oveq1 5881 . . . . . 6  |-  ( r  =  R  ->  (
r ^ r n )  =  ( R ^ r n ) )
65iuneq2d 3946 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
7 eqid 2296 . . . . 5  |-  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  =  ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) )
86, 7fvmptg 5616 . . . 4  |-  ( ( R  e.  _V  /\  U_ n  e.  NN0  ( R ^ r n )  e.  _V )  -> 
( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n ) )
91, 4, 8sylancl 643 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n ) )
10 breq 4041 . . . 4  |-  ( ( ( r  e.  _V  |->  U_ n  e.  NN0  (
r ^ r n ) ) `  R
)  =  U_ n  e.  NN0  ( R ^
r n )  -> 
( A ( ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) B  <->  A U_ n  e. 
NN0  ( R ^
r n ) B ) )
11 eliun 3925 . . . . . 6  |-  ( <. A ,  B >.  e. 
U_ n  e.  NN0  ( R ^ r n )  <->  E. n  e.  NN0  <. A ,  B >.  e.  ( R ^ r n ) )
1211a1i 10 . . . . 5  |-  ( ph  ->  ( <. A ,  B >.  e.  U_ n  e. 
NN0  ( R ^
r n )  <->  E. n  e.  NN0  <. A ,  B >.  e.  ( R ^
r n ) ) )
13 df-br 4040 . . . . 5  |-  ( A
U_ n  e.  NN0  ( R ^ r n ) B  <->  <. A ,  B >.  e.  U_ n  e.  NN0  ( R ^
r n ) )
14 df-br 4040 . . . . . 6  |-  ( A ( R ^ r n ) B  <->  <. A ,  B >.  e.  ( R ^ r n ) )
1514rexbii 2581 . . . . 5  |-  ( E. n  e.  NN0  A
( R ^ r n ) B  <->  E. n  e.  NN0  <. A ,  B >.  e.  ( R ^
r n ) )
1612, 13, 153bitr4g 279 . . . 4  |-  ( ph  ->  ( A U_ n  e.  NN0  ( R ^
r n ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
1710, 16sylan9bb 680 . . 3  |-  ( ( ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n )  /\  ph )  -> 
( A ( ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
189, 17mpancom 650 . 2  |-  ( ph  ->  ( A ( ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
19 df-rtrclrec 24054 . . 3  |-  t
*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )
20 fveq1 5540 . . . . . 6  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( t *rec
`  R )  =  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) )
2120breqd 4050 . . . . 5  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( A ( t *rec `  R
) B  <->  A (
( r  e.  _V  |->  U_ n  e.  NN0  (
r ^ r n ) ) `  R
) B ) )
2221bibi1d 310 . . . 4  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( ( A ( t *rec `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B )  <->  ( A
( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) ) )
2322imbi2d 307 . . 3  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( ( ph  ->  ( A ( t *rec `  R ) B 
<->  E. n  e.  NN0  A ( R ^ r n ) B ) )  <->  ( ph  ->  ( A ( ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) ) `
 R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) ) ) )
2419, 23ax-mp 8 . 2  |-  ( (
ph  ->  ( A ( t *rec `  R
) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )  <->  ( ph  ->  ( A ( ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) ) )
2518, 24mpbir 200 1  |-  ( ph  ->  ( A ( t *rec `  R ) B 
<->  E. n  e.  NN0  A ( R ^ r n ) B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   <.cop 3656   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   NN0cn0 9981   ^ rcrelexp 24038   t *reccrtrcl 24053
This theorem is referenced by:  rtrclreclem.trans  24058  rtrclind  24061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-n0 9982  df-rtrclrec 24054
  Copyright terms: Public domain W3C validator