Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trpredeq2 Structured version   Unicode version

Theorem trpredeq2 29231
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 29174 . . . . . . 7  |-  ( A  =  B  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  B , 
y ) )
21iuneq2d 4358 . . . . . 6  |-  ( A  =  B  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( R ,  B ,  y ) )
32mpteq2dv 4540 . . . . 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 29174 . . . . 5  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
5 rdgeq12 7091 . . . . . 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 5278 . . . . 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 661 . . . 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 5236 . . 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 4261 . 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 29228 . 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 29228 . 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 2533 1  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   _Vcvv 3118   U.cuni 4251   U_ciun 4331    |-> cmpt 4511   ran crn 5006    |` cres 5007   omcom 6695   reccrdg 7087   Predcpred 29170   TrPredctrpred 29227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fv 5602  df-recs 7054  df-rdg 7088  df-pred 29171  df-trpred 29228
This theorem is referenced by:  trpredeq2d  29234
  Copyright terms: Public domain W3C validator