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Theorem bnj1124 29294
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 228 . 2  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 biid 228 . 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 228 . 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 228 . 2  |-  ( ( i  e.  n  /\  z  e.  ( f `  i ) )  <->  ( i  e.  n  /\  z  e.  ( f `  i
) ) )
7 eqid 2435 . 2  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
8 eqid 2435 . 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 228 . 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 228 . 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 228 . 2  |-  ( [. j  /  i ]. (
f `  (/) )  = 
pred ( X ,  A ,  R )  <->  [. j  /  i ]. ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
12 biid 228 . 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 228 . 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 228 . 2  |-  ( [. j  /  i ]. th  <->  [. j  /  i ]. th )
15 biid 228 . 2  |-  ( [. j  /  i ]. ta  <->  [. j  /  i ]. ta )
16 biid 228 . 2  |-  ( [. j  /  i ]. (
i  e.  n  /\  z  e.  ( f `  i ) )  <->  [. j  / 
i ]. ( i  e.  n  /\  z  e.  ( f `  i
) ) )
17 biid 228 . 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 228 . 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 228 . 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 29293 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948   [.wsbc 3153    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   U_ciun 4085   class class class wbr 4204    _E cep 4484   suc csuc 4575   omcom 4837   dom cdm 4870    Fn wfn 5441   ` cfv 5446    /\ w-bnj17 28987    predc-bnj14 28989    FrSe w-bnj15 28993    trClc-bnj18 28995    TrFow-bnj19 28997
This theorem is referenced by:  bnj1125  29298  bnj1136  29303  bnj1408  29342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-iota 5410  df-fn 5449  df-fv 5454  df-bnj17 28988  df-bnj18 28996  df-bnj19 28998
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