Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trpredeq1 Structured version   Unicode version

Theorem trpredeq1 25503
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 25446 . . . . . . . 8  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
21iuneq2d 4120 . . . . . . 7  |-  ( R  =  S  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( S ,  A ,  y ) )
32mpteq2dv 4299 . . . . . 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 25446 . . . . . 6  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
5 rdgeq12 6674 . . . . . 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 644 . . . . 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 5148 . . . 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 5100 . . 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 4028 . 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 25501 . 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 25501 . 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 2495 1  |-  ( R  =  S  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( S ,  A ,  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2958   U.cuni 4017   U_ciun 4095    e. cmpt 4269   omcom 4848   ran crn 4882    |` cres 4883   reccrdg 6670   Predcpred 25443   TrPredctrpred 25500
This theorem is referenced by:  trpredeq1d  25506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-recs 6636  df-rdg 6671  df-pred 25444  df-trpred 25501
  Copyright terms: Public domain W3C validator