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Theorem dfrtrclrec2 29241
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 10822 . . . . 5  |-  NN0  e.  _V
3 ovex 6324 . . . . 5  |-  ( R ^r n )  e.  _V
42, 3iunex 6779 . . . 4  |-  U_ n  e.  NN0  ( R ^r n )  e. 
_V
5 oveq1 6303 . . . . . 6  |-  ( r  =  R  ->  (
r ^r n )  =  ( R ^r n ) )
65iuneq2d 4359 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^r n )  = 
U_ n  e.  NN0  ( R ^r n ) )
7 eqid 2457 . . . . 5  |-  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )  =  ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^r n ) )
86, 7fvmptg 5954 . . . 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 4458 . . . 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 4337 . . . . . 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 4457 . . . . 5  |-  ( A
U_ n  e.  NN0  ( R ^r n ) B  <->  <. A ,  B >.  e.  U_ n  e.  NN0  ( R ^r n ) )
14 df-br 4457 . . . . . 6  |-  ( A ( R ^r n ) B  <->  <. A ,  B >.  e.  ( R ^r n ) )
1514rexbii 2959 . . . . 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 29240 . . 3  |-  t*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )
20 fveq1 5871 . . . . . 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 4467 . . . . 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 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109   <.cop 4038   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515   Rel wrel 5013   ` cfv 5594  (class class class)co 6296   NN0cn0 10816   ^rcrelexp 29225   t*reccrtrcl 29239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-om 6700  df-recs 7060  df-rdg 7094  df-nn 10557  df-n0 10817  df-rtrclrec 29240
This theorem is referenced by:  rtrclreclem.trans  29244  rtrclind  29247
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