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Theorem trpredeq2 29469
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 29412 . . . . . . 7  |-  ( A  =  B  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  B , 
y ) )
21iuneq2d 4270 . . . . . 6  |-  ( A  =  B  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( R ,  B ,  y ) )
32mpteq2dv 4454 . . . . 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 29412 . . . . 5  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
5 rdgeq12 6997 . . . . . 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 5185 . . . . 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 659 . . . 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 5143 . . 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 4173 . 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 29466 . 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 29466 . 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 2448 1  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   _Vcvv 3034   U.cuni 4163   U_ciun 4243    |-> cmpt 4425   ran crn 4914    |` cres 4915   omcom 6599   reccrdg 6993   Predcpred 29408   TrPredctrpred 29465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-xp 4919  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fv 5504  df-recs 6960  df-rdg 6994  df-pred 29409  df-trpred 29466
This theorem is referenced by:  trpredeq2d  29472
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