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Theorem trpredeq1 24781
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 24727 . . . . . . . 8  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
21iuneq2d 4009 . . . . . . 7  |-  ( R  =  S  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( S ,  A ,  y ) )
32mpteq2dv 4186 . . . . . 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 24727 . . . . . 6  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
5 rdgeq12 6510 . . . . . 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 642 . . . . 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 5033 . . . 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 4985 . . 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 3917 . 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 24779 . 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 24779 . 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 2415 1  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   _Vcvv 2864   U.cuni 3906   U_ciun 3984    e. cmpt 4156   omcom 4735   ran crn 4769    |` cres 4770   reccrdg 6506   Predcpred 24725   TrPredctrpred 24778
This theorem is referenced by:  trpredeq1d  24784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fv 5342  df-recs 6472  df-rdg 6507  df-pred 24726  df-trpred 24779
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