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Theorem bnj1318 32014
Description: Technical lemma for bnj60 32051. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1318  |-  ( X  =  Y  ->  trCl ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )

Proof of Theorem bnj1318
Dummy variables  f 
i  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj602 31906 . . . . . . 7  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
21eqeq2d 2453 . . . . . 6  |-  ( X  =  Y  ->  (
( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( Y ,  A ,  R )
) )
323anbi2d 1294 . . . . 5  |-  ( X  =  Y  ->  (
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )  <-> 
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
43rexbidv 2735 . . . 4  |-  ( X  =  Y  ->  ( E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )  <->  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
54abbidv 2556 . . 3  |-  ( X  =  Y  ->  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  =  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } )
6 hbab1 2431 . . . 4  |-  ( z  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  ->  A. f 
z  e.  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } )
7 hbab1 2431 . . . 4  |-  ( z  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  ->  A. f 
z  e.  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } )
86, 7bnj1316 31812 . . 3  |-  ( { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  =  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  ->  U_ f  e. 
{ f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )  = 
U_ f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i ) )
95, 8syl 16 . 2  |-  ( X  =  Y  ->  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )  = 
U_ f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i ) )
10 biid 236 . . 3  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
11 biid 236 . . 3  |-  ( A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
12 eqid 2442 . . 3  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
13 eqid 2442 . . 3  |-  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  =  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }
1410, 11, 12, 13bnj882 31917 . 2  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )
15 biid 236 . . 3  |-  ( ( f `  (/) )  = 
pred ( Y ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( Y ,  A ,  R )
)
16 eqid 2442 . . 3  |-  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  =  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }
1715, 11, 12, 16bnj882 31917 . 2  |-  trCl ( Y ,  A ,  R )  =  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )
189, 14, 173eqtr4g 2499 1  |-  ( X  =  Y  ->  trCl ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2428   A.wral 2714   E.wrex 2715    \ cdif 3324   (/)c0 3636   {csn 3876   U_ciun 4170   suc csuc 4720   dom cdm 4839    Fn wfn 5412   ` cfv 5417   omcom 6475    predc-bnj14 31674    trClc-bnj18 31680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-iun 4172  df-br 4292  df-bnj14 31675  df-bnj18 31681
This theorem is referenced by:  bnj1373  32019  bnj1408  32025  bnj1417  32030  bnj1489  32045
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