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Theorem dfrtrclrec2 27343
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 10583 . . . . 5  |-  NN0  e.  _V
3 ovex 6114 . . . . 5  |-  ( R ^r n )  e.  _V
42, 3iunex 6555 . . . 4  |-  U_ n  e.  NN0  ( R ^r n )  e. 
_V
5 oveq1 6096 . . . . . 6  |-  ( r  =  R  ->  (
r ^r n )  =  ( R ^r n ) )
65iuneq2d 4195 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^r n )  = 
U_ n  e.  NN0  ( R ^r n ) )
7 eqid 2441 . . . . 5  |-  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )  =  ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^r n ) )
86, 7fvmptg 5770 . . . 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 662 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^r n ) )
10 breq 4292 . . . 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 4173 . . . . . 6  |-  ( <. A ,  B >.  e. 
U_ n  e.  NN0  ( R ^r n )  <->  E. n  e.  NN0  <. A ,  B >.  e.  ( R ^r n ) )
1211a1i 11 . . . . 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 4291 . . . . 5  |-  ( A
U_ n  e.  NN0  ( R ^r n ) B  <->  <. A ,  B >.  e.  U_ n  e.  NN0  ( R ^r n ) )
14 df-br 4291 . . . . . 6  |-  ( A ( R ^r n ) B  <->  <. A ,  B >.  e.  ( R ^r n ) )
1514rexbii 2738 . . . . 5  |-  ( E. n  e.  NN0  A
( R ^r n ) B  <->  E. n  e.  NN0  <. A ,  B >.  e.  ( R ^r n ) )
1612, 13, 153bitr4g 288 . . . 4  |-  ( ph  ->  ( A U_ n  e.  NN0  ( R ^r n ) B  <->  E. n  e.  NN0  A ( R ^r n ) B ) )
1710, 16sylan9bb 699 . . 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 669 . 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 27342 . . 3  |-  t*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )
20 fveq1 5688 . . . . . 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 4301 . . . . 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 319 . . . 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 316 . . 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 5 . 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 209 1  |-  ( ph  ->  ( A ( t*rec `  R ) B 
<->  E. n  e.  NN0  A ( R ^r n ) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   E.wrex 2714   _Vcvv 2970   <.cop 3881   U_ciun 4169   class class class wbr 4290    e. cmpt 4348   Rel wrel 4843   ` cfv 5416  (class class class)co 6089   NN0cn0 10577   ^rcrelexp 27327   t*reccrtrcl 27341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-i2m1 9348  ax-1ne0 9349  ax-rrecex 9352  ax-cnre 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-om 6475  df-recs 6830  df-rdg 6864  df-nn 10321  df-n0 10578  df-rtrclrec 27342
This theorem is referenced by:  rtrclreclem.trans  27346  rtrclind  27349
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