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Theorem trpredeq1 25437
Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
trpredeq1  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )

Proof of Theorem trpredeq1
Dummy variables  a 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq1 25383 . . . . . . . 8  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
21iuneq2d 4078 . . . . . . 7  |-  ( R  =  S  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( S ,  A ,  y ) )
32mpteq2dv 4256 . . . . . 6  |-  ( R  =  S  ->  (
a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A , 
y ) )  =  ( a  e.  _V  |->  U_ y  e.  a  Pred ( S ,  A , 
y ) ) )
4 predeq1 25383 . . . . . 6  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
5 rdgeq12 6630 . . . . . 6  |-  ( ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A , 
y ) )  =  ( a  e.  _V  |->  U_ y  e.  a  Pred ( S ,  A , 
y ) )  /\  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )  ->  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 ( S ,  A ,  y ) ) ,  Pred ( S ,  A ,  X ) ) )
63, 4, 5syl2anc 643 . . . . 5  |-  ( R  =  S  ->  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 ( S ,  A ,  y ) ) ,  Pred ( S ,  A ,  X ) ) )
76reseq1d 5104 . . . 4  |-  ( R  =  S  ->  ( 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 ( S ,  A ,  y )
) ,  Pred ( S ,  A ,  X ) )  |`  om ) )
87rneqd 5056 . . 3  |-  ( R  =  S  ->  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 ( S ,  A ,  y ) ) ,  Pred ( S ,  A ,  X ) )  |`  om ) )
98unieqd 3986 . 2  |-  ( R  =  S  ->  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 ( S ,  A , 
y ) ) , 
Pred ( S ,  A ,  X )
)  |`  om ) )
10 df-trpred 25435 . 2  |-  TrPred ( R ,  A ,  X
)  =  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) ,  Pred ( R ,  A ,  X ) )  |`  om )
11 df-trpred 25435 . 2  |-  TrPred ( S ,  A ,  X
)  =  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( S ,  A ,  y ) ) ,  Pred ( S ,  A ,  X ) )  |`  om )
129, 10, 113eqtr4g 2461 1  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2916   U.cuni 3975   U_ciun 4053    e. cmpt 4226   omcom 4804   ran crn 4838    |` cres 4839   reccrdg 6626   Predcpred 25381   TrPredctrpred 25434
This theorem is referenced by:  trpredeq1d  25440
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-recs 6592  df-rdg 6627  df-pred 25382  df-trpred 25435
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