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Theorem trpredeq1 25441
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 25387 . . . . . . . 8  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
21iuneq2d 4082 . . . . . . 7  |-  ( R  =  S  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( S ,  A ,  y ) )
32mpteq2dv 4260 . . . . . 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 25387 . . . . . 6  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
5 rdgeq12 6634 . . . . . 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 5108 . . . 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 5060 . . 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 3990 . 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 25439 . 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 25439 . 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 2465 1  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2920   U.cuni 3979   U_ciun 4057    e. cmpt 4230   omcom 4808   ran crn 4842    |` cres 4843   reccrdg 6630   Predcpred 25385   TrPredctrpred 25438
This theorem is referenced by:  trpredeq1d  25444
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 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fv 5425  df-recs 6596  df-rdg 6631  df-pred 25386  df-trpred 25439
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