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Theorem dfrtrclrec2 24040
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 9971 . . . . 5  |-  NN0  e.  _V
3 ovex 5883 . . . . 5  |-  ( R ^ r n )  e.  _V
42, 3iunex 5770 . . . 4  |-  U_ n  e.  NN0  ( R ^
r n )  e. 
_V
5 oveq1 5865 . . . . . 6  |-  ( r  =  R  ->  (
r ^ r n )  =  ( R ^ r n ) )
65iuneq2d 3930 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
7 eqid 2283 . . . . 5  |-  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  =  ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) )
86, 7fvmptg 5600 . . . 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 4025 . . . 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 3909 . . . . . 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 4024 . . . . 5  |-  ( A
U_ n  e.  NN0  ( R ^ r n ) B  <->  <. A ,  B >.  e.  U_ n  e.  NN0  ( R ^
r n ) )
14 df-br 4024 . . . . . 6  |-  ( A ( R ^ r n ) B  <->  <. A ,  B >.  e.  ( R ^ r n ) )
1514rexbii 2568 . . . . 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 24039 . . 3  |-  t
*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )
20 fveq1 5524 . . . . . 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 4034 . . . . 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 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   <.cop 3643   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   NN0cn0 9965   ^ rcrelexp 24023   t *reccrtrcl 24038
This theorem is referenced by:  rtrclreclem.trans  24043  rtrclind  24046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-rtrclrec 24039
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