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Theorem dfrtrclrec2 28538
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 10797 . . . . 5  |-  NN0  e.  _V
3 ovex 6307 . . . . 5  |-  ( R ^r n )  e.  _V
42, 3iunex 6761 . . . 4  |-  U_ n  e.  NN0  ( R ^r n )  e. 
_V
5 oveq1 6289 . . . . . 6  |-  ( r  =  R  ->  (
r ^r n )  =  ( R ^r n ) )
65iuneq2d 4352 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^r n )  = 
U_ n  e.  NN0  ( R ^r n ) )
7 eqid 2467 . . . . 5  |-  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )  =  ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^r n ) )
86, 7fvmptg 5946 . . . 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 4449 . . . 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 4330 . . . . . 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 4448 . . . . 5  |-  ( A
U_ n  e.  NN0  ( R ^r n ) B  <->  <. A ,  B >.  e.  U_ n  e.  NN0  ( R ^r n ) )
14 df-br 4448 . . . . . 6  |-  ( A ( R ^r n ) B  <->  <. A ,  B >.  e.  ( R ^r n ) )
1514rexbii 2965 . . . . 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 28537 . . 3  |-  t*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )
20 fveq1 5863 . . . . . 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 4458 . . . . 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 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   <.cop 4033   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   NN0cn0 10791   ^rcrelexp 28522   t*reccrtrcl 28536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533  df-n0 10792  df-rtrclrec 28537
This theorem is referenced by:  rtrclreclem.trans  28541  rtrclind  28544
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