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Theorem trpredeq2 23635
Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
trpredeq2  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )

Proof of Theorem trpredeq2
Dummy variables  a 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq2 23581 . . . . . . 7  |-  ( A  =  B  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  B , 
y ) )
21iuneq2d 3930 . . . . . 6  |-  ( A  =  B  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( R ,  B ,  y ) )
32mpteq2dv 4107 . . . . 5  |-  ( A  =  B  ->  (
a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A , 
y ) )  =  ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  B , 
y ) ) )
4 predeq2 23581 . . . . 5  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
5 rdgeq12 6426 . . . . . 6  |-  ( ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A , 
y ) )  =  ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  B , 
y ) )  /\  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  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 ( R ,  B ,  y ) ) ,  Pred ( R ,  B ,  X ) ) )
65reseq1d 4954 . . . . 5  |-  ( ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A , 
y ) )  =  ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  B , 
y ) )  /\  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )  -> 
( 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 ,  B , 
y ) ) , 
Pred ( R ,  B ,  X )
)  |`  om ) )
73, 4, 6syl2anc 642 . . . 4  |-  ( A  =  B  ->  ( 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 ,  B ,  y )
) ,  Pred ( R ,  B ,  X ) )  |`  om ) )
87rneqd 4906 . . 3  |-  ( A  =  B  ->  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 ,  B ,  y ) ) ,  Pred ( R ,  B ,  X ) )  |`  om ) )
98unieqd 3838 . 2  |-  ( A  =  B  ->  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 ,  B , 
y ) ) , 
Pred ( R ,  B ,  X )
)  |`  om ) )
10 df-trpred 23632 . 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 23632 . 2  |-  TrPred ( R ,  B ,  X
)  =  U. ran  ( rec ( ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  B ,  y ) ) ,  Pred ( R ,  B ,  X ) )  |`  om )
129, 10, 113eqtr4g 2340 1  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623   _Vcvv 2788   U.cuni 3827   U_ciun 3905    e. cmpt 4077   omcom 4656   ran crn 4690    |` cres 4691   reccrdg 6422   Predcpred 23578   TrPredctrpred 23631
This theorem is referenced by:  trpredeq2d  23638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fv 5263  df-recs 6388  df-rdg 6423  df-pred 23579  df-trpred 23632
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