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Theorem dfrtrclrec2 25143
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 10227 . . . . 5  |-  NN0  e.  _V
3 ovex 6106 . . . . 5  |-  ( R ^ r n )  e.  _V
42, 3iunex 5991 . . . 4  |-  U_ n  e.  NN0  ( R ^
r n )  e. 
_V
5 oveq1 6088 . . . . . 6  |-  ( r  =  R  ->  (
r ^ r n )  =  ( R ^ r n ) )
65iuneq2d 4118 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
7 eqid 2436 . . . . 5  |-  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  =  ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) )
86, 7fvmptg 5804 . . . 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 644 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n ) )
10 breq 4214 . . . 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 4097 . . . . . 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 4213 . . . . 5  |-  ( A
U_ n  e.  NN0  ( R ^ r n ) B  <->  <. A ,  B >.  e.  U_ n  e.  NN0  ( R ^
r n ) )
14 df-br 4213 . . . . . 6  |-  ( A ( R ^ r n ) B  <->  <. A ,  B >.  e.  ( R ^ r n ) )
1514rexbii 2730 . . . . 5  |-  ( E. n  e.  NN0  A
( R ^ r n ) B  <->  E. n  e.  NN0  <. A ,  B >.  e.  ( R ^
r n ) )
1612, 13, 153bitr4g 280 . . . 4  |-  ( ph  ->  ( A U_ n  e.  NN0  ( R ^
r n ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
1710, 16sylan9bb 681 . . 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 651 . 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 25142 . . 3  |-  t
*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )
20 fveq1 5727 . . . . . 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 4223 . . . . 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 311 . . . 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 308 . . 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 201 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 177    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956   <.cop 3817   U_ciun 4093   class class class wbr 4212    e. cmpt 4266   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   NN0cn0 10221   ^ rcrelexp 25127   t *reccrtrcl 25141
This theorem is referenced by:  rtrclreclem.trans  25146  rtrclind  25149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-recs 6633  df-rdg 6668  df-nn 10001  df-n0 10222  df-rtrclrec 25142
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