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Theorem trpredeq3 25249
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 25195 . . . . . 6  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
2 rdgeq2 6606 . . . . . 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 5085 . . . 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 5037 . . 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 3968 . 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 25245 . 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 25245 . 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 2444 1  |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  A ,  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2899   U.cuni 3957   U_ciun 4035    e. cmpt 4207   omcom 4785   ran crn 4819    |` cres 4820   reccrdg 6603   Predcpred 25191   TrPredctrpred 25244
This theorem is referenced by:  trpredeq3d  25252  dftrpred3g  25260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fv 5402  df-recs 6569  df-rdg 6604  df-pred 25192  df-trpred 25245
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