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Theorem trpredeq2 30249
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 5402 . . . . . . 7  |-  ( A  =  B  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  B , 
y ) )
21iuneq2d 4329 . . . . . 6  |-  ( A  =  B  ->  U_ y  e.  a  Pred ( R ,  A ,  y )  =  U_ y  e.  a  Pred ( R ,  B ,  y ) )
32mpteq2dv 4513 . . . . 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 5402 . . . . 5  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
5 rdgeq12 7139 . . . . . 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 5124 . . . . 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 665 . . . 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 5082 . . 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 4232 . 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 30246 . 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 30246 . 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 2495 1  |-  ( A  =  B  ->  TrPred ( R ,  A ,  X
)  =  TrPred ( R ,  B ,  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   _Vcvv 3087   U.cuni 4222   U_ciun 4302    |-> cmpt 4484   ran crn 4855    |` cres 4856   Predcpred 5398   omcom 6706   reccrdg 7135   TrPredctrpred 30245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-iota 5565  df-fv 5609  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-trpred 30246
This theorem is referenced by:  trpredeq2d  30252
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