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Theorem bnj906 28962
Description: Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj906  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)

Proof of Theorem bnj906
Dummy variables  f 
i  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1onn 6637 . . . . . . . 8  |-  1o  e.  om
2 1n0 6494 . . . . . . . 8  |-  1o  =/=  (/)
3 eldifsn 3749 . . . . . . . 8  |-  ( 1o  e.  ( om  \  { (/)
} )  <->  ( 1o  e.  om  /\  1o  =/=  (/) ) )
41, 2, 3mpbir2an 886 . . . . . . 7  |-  1o  e.  ( om  \  { (/) } )
5 ne0i 3461 . . . . . . 7  |-  ( 1o  e.  ( om  \  { (/)
} )  ->  ( om  \  { (/) } )  =/=  (/) )
64, 5ax-mp 8 . . . . . 6  |-  ( om 
\  { (/) } )  =/=  (/)
7 biid 227 . . . . . . 7  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
8 biid 227 . . . . . . 7  |-  ( 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 ) ) )
9 eqid 2283 . . . . . . 7  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
107, 8, 9bnj852 28953 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  ( om  \  { (/) } ) E! f ( 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 r19.2z 3543 . . . . . 6  |-  ( ( ( om  \  { (/)
} )  =/=  (/)  /\  A. n  e.  ( om  \  { (/) } ) E! f ( 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  \  { (/)
} ) E! f ( 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 ) ) ) )
126, 10, 11sylancr 644 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. n  e.  ( om  \  { (/) } ) E! f ( 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 ) ) ) )
13 euex 2166 . . . . 5  |-  ( E! f ( 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. f ( 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 ) ) ) )
1412, 13bnj31 28745 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. n  e.  ( om  \  { (/) } ) E. f ( 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 ) ) ) )
15 rexcom4 2807 . . . 4  |-  ( E. n  e.  ( om 
\  { (/) } ) E. f ( 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. 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 ) ) ) )
1614, 15sylib 188 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  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 ) ) ) )
17 abid 2271 . . 3  |-  ( 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 ) ) ) }  <->  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 ) ) ) )
1816, 17bnj1198 28828 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. f  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 ) ) ) } )
19 simp2 956 . . . . . . 7  |-  ( ( 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 `  (/) )  =  pred ( X ,  A ,  R ) )
2019reximi 2650 . . . . . 6  |-  ( 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 `  (/) )  =  pred ( X ,  A ,  R ) )
2117, 20sylbi 187 . . . . 5  |-  ( 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 ) ) ) }  ->  E. n  e.  ( om  \  { (/)
} ) ( f `
 (/) )  =  pred ( X ,  A ,  R ) )
22 df-rex 2549 . . . . . 6  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
) )
23 19.41v 1842 . . . . . . 7  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  <->  ( E. n  n  e.  ( om  \  { (/) } )  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
2423simprbi 450 . . . . . 6  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  (/) )  =  pred ( X ,  A ,  R ) )
2522, 24sylbi 187 . . . . 5  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2621, 25syl 15 . . . 4  |-  ( 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 ) ) ) }  ->  ( f `  (/) )  =  pred ( X ,  A ,  R ) )
27 eqid 2283 . . . . . . 7  |-  { 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 ) ) ) }
289, 27bnj900 28961 . . . . . 6  |-  ( 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 ) ) ) }  ->  (/)  e.  dom  f )
29 fveq2 5525 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
3029ssiun2s 3946 . . . . . 6  |-  ( (/)  e.  dom  f  ->  (
f `  (/) )  C_  U_ i  e.  dom  f
( f `  i
) )
3128, 30syl 15 . . . . 5  |-  ( 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 ) ) ) }  ->  ( f `  (/) )  C_  U_ i  e.  dom  f ( f `
 i ) )
32 ssiun2 3945 . . . . . 6  |-  ( 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 )  C_  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 ) )
337, 8, 9, 27bnj882 28958 . . . . . 6  |-  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 )
3432, 33syl6sseqr 3225 . . . . 5  |-  ( 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 )  C_  trCl ( X ,  A ,  R ) )
3531, 34sstrd 3189 . . . 4  |-  ( 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 ) ) ) }  ->  ( f `  (/) )  C_  trCl ( X ,  A ,  R ) )
3626, 35eqsstr3d 3213 . . 3  |-  ( 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 ) ) ) }  ->  pred ( X ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) )
3736exlimiv 1666 . 2  |-  ( E. f  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 ) ) ) }  ->  pred ( X ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) )
3818, 37syl 15 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   U_ciun 3905   suc csuc 4394   omcom 4656   dom cdm 4689    Fn wfn 5250   ` cfv 5255   1oc1o 6472    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1137  29025  bnj1136  29027  bnj1175  29034  bnj1177  29036  bnj1413  29065  bnj1408  29066  bnj1417  29071  bnj1442  29079  bnj1452  29082
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
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-reu 2550  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-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720
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