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Theorem trpredeq3 25492
Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
trpredeq3  |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  A ,  Y
) )

Proof of Theorem trpredeq3
Dummy variables  a 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq3 25435 . . . . . 6  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
2 rdgeq2 6662 . . . . . 6  |-  ( Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y )  ->  rec ( ( a  e. 
_V  |->  U_ y  e.  a 
Pred ( R ,  A ,  y )
) ,  Pred ( R ,  A ,  X ) )  =  rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  Y ) ) )
31, 2syl 16 . . . . 5  |-  ( X  =  Y  ->  rec ( ( a  e. 
_V  |->  U_ y  e.  a 
Pred ( R ,  A ,  y )
) ,  Pred ( R ,  A ,  X ) )  =  rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  Y ) ) )
43reseq1d 5137 . . . 4  |-  ( X  =  Y  ->  ( rec ( ( a  e. 
_V  |->  U_ y  e.  a 
Pred ( R ,  A ,  y )
) ,  Pred ( R ,  A ,  X ) )  |`  om )  =  ( rec ( ( a  e. 
_V  |->  U_ y  e.  a 
Pred ( R ,  A ,  y )
) ,  Pred ( R ,  A ,  Y ) )  |`  om ) )
54rneqd 5089 . . 3  |-  ( X  =  Y  ->  ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  X ) )  |`  om )  =  ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  Y ) )  |`  om ) )
65unieqd 4018 . 2  |-  ( X  =  Y  ->  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  X ) )  |`  om )  =  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A , 
y ) ) , 
Pred ( R ,  A ,  Y )
)  |`  om ) )
7 df-trpred 25488 . 2  |-  TrPred ( R ,  A ,  X
)  =  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  X ) )  |`  om )
8 df-trpred 25488 . 2  |-  TrPred ( R ,  A ,  Y
)  =  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  Y ) )  |`  om )
96, 7, 83eqtr4g 2492 1  |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  A ,  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   _Vcvv 2948   U.cuni 4007   U_ciun 4085    e. cmpt 4258   omcom 4837   ran crn 4871    |` cres 4872   reccrdg 6659   Predcpred 25430   TrPredctrpred 25487
This theorem is referenced by:  trpredeq3d  25495  dftrpred3g  25503
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-recs 6625  df-rdg 6660  df-pred 25431  df-trpred 25488
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