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Theorem bnj1124 29018
Description: Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1124.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1124.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
Assertion
Ref Expression
bnj1124  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )

Proof of Theorem bnj1124
Dummy variables  f 
i  j  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 227 . 2  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 biid 227 . 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 biid 227 . 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 ) ) ) )
4 bnj1124.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1124.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 biid 227 . 2  |-  ( ( i  e.  n  /\  z  e.  ( f `  i ) )  <->  ( i  e.  n  /\  z  e.  ( f `  i
) ) )
7 eqid 2283 . 2  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
8 eqid 2283 . 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 ) ) ) }
9 biid 227 . 2  |-  ( ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )  <->  ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)
10 biid 227 . 2  |-  ( A. j  e.  n  (
j  _E  i  ->  [. j  /  i ]. ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
) )
11 biid 227 . 2  |-  ( [. j  /  i ]. (
f `  (/) )  = 
pred ( X ,  A ,  R )  <->  [. j  /  i ]. ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
12 biid 227 . 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 ) ) )
13 biid 227 . 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 ) ) ) )
14 biid 227 . 2  |-  ( [. j  /  i ]. th  <->  [. j  /  i ]. th )
15 biid 227 . 2  |-  ( [. j  /  i ]. ta  <->  [. j  /  i ]. ta )
16 biid 227 . 2  |-  ( [. j  /  i ]. (
i  e.  n  /\  z  e.  ( f `  i ) )  <->  [. j  / 
i ]. ( i  e.  n  /\  z  e.  ( f `  i
) ) )
17 biid 227 . 2  |-  ( [. j  /  i ]. (
( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )  <->  [. j  /  i ]. ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)
18 biid 227 . 2  |-  ( ( ( j  e.  n  /\  j  _E  i
)  ->  [. j  / 
i ]. ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)  <->  ( ( j  e.  n  /\  j  _E  i )  ->  [. j  /  i ]. (
( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
) )
19 biid 227 . 2  |-  ( ( i  e.  n  /\  ( ( j  e.  n  /\  j  _E  i )  ->  [. j  /  i ]. (
( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)  /\  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 ) ) ) }  /\  i  e. 
dom  f )  <->  ( i  e.  n  /\  (
( j  e.  n  /\  j  _E  i
)  ->  [. j  / 
i ]. ( ( 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 ) ) ) }  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)  /\  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 ) ) ) }  /\  i  e. 
dom  f ) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19bnj1030 29017 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656   dom cdm 4689    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem is referenced by:  bnj1125  29022  bnj1136  29027  bnj1408  29066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fn 5258  df-fv 5263  df-bnj17 28712  df-bnj18 28720  df-bnj19 28722
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