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Theorem bnj1127 34467
Description: Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1127  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )

Proof of Theorem bnj1127
Dummy variables  f 
i  j  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 236 . 2  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 biid 236 . 2  |-  ( 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 ) ) )
3 eqid 2454 . 2  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
4 eqid 2454 . 2  |-  { 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 ) ) ) }
5 biid 236 . 2  |-  ( ( 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 ) ) )  <-> 
( 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 ) ) ) )
6 biid 236 . 2  |-  ( ( ( 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 `  i )  C_  A
)  <->  ( ( 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 `  i )  C_  A
) )
7 biid 236 . 2  |-  ( A. j  e.  n  (
j  _E  i  ->  [. j  /  i ]. ( ( 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 `  i )  C_  A
) )  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. (
( 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 `  i )  C_  A
) ) )
8 biid 236 . 2  |-  ( [. j  /  i ]. (
f `  (/) )  = 
pred ( X ,  A ,  R )  <->  [. j  /  i ]. ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
9 biid 236 . 2  |-  ( [. j  /  i ]. A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  [. j  / 
i ]. A. i  e. 
om  ( suc  i  e.  n  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
10 biid 236 . 2  |-  ( [. j  /  i ]. (
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 ) ) )  <->  [. j  /  i ]. ( 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 ) ) ) )
11 biid 236 . 2  |-  ( [. j  /  i ]. (
( 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 `  i )  C_  A
)  <->  [. j  /  i ]. ( ( 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 `  i )  C_  A
) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1128 34466 1  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805   [.wsbc 3324    \ cdif 3458    C_ wss 3461   (/)c0 3783   {csn 4016   U_ciun 4315   class class class wbr 4439    _E cep 4778   suc csuc 4869    Fn wfn 5565   ` cfv 5570   omcom 6673    /\ w-bnj17 34158    predc-bnj14 34160    trClc-bnj18 34166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-iota 5534  df-fn 5573  df-fv 5578  df-om 6674  df-bnj17 34159  df-bnj14 34161  df-bnj18 34167
This theorem is referenced by:  bnj1125  34468  bnj1136  34473  bnj1413  34511
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