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Theorem bnj1127 32334
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 32333 1  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800   [.wsbc 3294    \ cdif 3436    C_ wss 3439   (/)c0 3748   {csn 3988   U_ciun 4282   class class class wbr 4403    _E cep 4741   suc csuc 4832    Fn wfn 5524   ` cfv 5529   omcom 6589    /\ w-bnj17 32026    predc-bnj14 32028    trClc-bnj18 32034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-iota 5492  df-fn 5532  df-fv 5537  df-om 6590  df-bnj17 32027  df-bnj14 32029  df-bnj18 32035
This theorem is referenced by:  bnj1125  32335  bnj1136  32340  bnj1413  32378
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