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Theorem trpredeq2 25500
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 25443 . . . . . . 7  |-  ( A  =  B  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  B , 
y ) )
21iuneq2d 4119 . . . . . 6  |-  ( A  =  B  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( R ,  B ,  y ) )
32mpteq2dv 4297 . . . . 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 25443 . . . . 5  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
5 rdgeq12 6672 . . . . . 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 5146 . . . . 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 644 . . . 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 5098 . . 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 4027 . 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 25497 . 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 25497 . 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 2494 1  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653   _Vcvv 2957   U.cuni 4016   U_ciun 4094    e. cmpt 4267   omcom 4846   ran crn 4880    |` cres 4881   reccrdg 6668   Predcpred 25439   TrPredctrpred 25496
This theorem is referenced by:  trpredeq2d  25503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fv 5463  df-recs 6634  df-rdg 6669  df-pred 25440  df-trpred 25497
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