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Theorem bnj906 29301
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 6882 . . . . . . . 8  |-  1o  e.  om
2 1n0 6739 . . . . . . . 8  |-  1o  =/=  (/)
3 eldifsn 3927 . . . . . . . 8  |-  ( 1o  e.  ( om  \  { (/)
} )  <->  ( 1o  e.  om  /\  1o  =/=  (/) ) )
41, 2, 3mpbir2an 887 . . . . . . 7  |-  1o  e.  ( om  \  { (/) } )
5 ne0i 3634 . . . . . . 7  |-  ( 1o  e.  ( om  \  { (/)
} )  ->  ( om  \  { (/) } )  =/=  (/) )
64, 5ax-mp 8 . . . . . 6  |-  ( om 
\  { (/) } )  =/=  (/)
7 biid 228 . . . . . . 7  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
8 biid 228 . . . . . . 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 2436 . . . . . . 7  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
107, 8, 9bnj852 29292 . . . . . 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 3717 . . . . . 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 645 . . . . 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 2304 . . . . 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 29084 . . . 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 2975 . . . 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 189 . . 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 2424 . . 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 29167 . 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 958 . . . . . . 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 2813 . . . . . 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 188 . . . . 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 2711 . . . . . 6  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
) )
23 19.41v 1924 . . . . . . 7  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  <->  ( E. n  n  e.  ( om  \  { (/) } )  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
2423simprbi 451 . . . . . 6  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  (/) )  =  pred ( X ,  A ,  R ) )
2522, 24sylbi 188 . . . . 5  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2621, 25syl 16 . . . 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 2436 . . . . . . 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 29300 . . . . . 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 5728 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
3029ssiun2s 4135 . . . . . 6  |-  ( (/)  e.  dom  f  ->  (
f `  (/) )  C_  U_ i  e.  dom  f
( f `  i
) )
3128, 30syl 16 . . . . 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 4134 . . . . . 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 29297 . . . . . 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 3395 . . . . 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 3358 . . . 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 3383 . . 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 1644 . 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 16 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 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706    \ cdif 3317    C_ wss 3320   (/)c0 3628   {csn 3814   U_ciun 4093   suc csuc 4583   omcom 4845   dom cdm 4878    Fn wfn 5449   ` cfv 5454   1oc1o 6717    predc-bnj14 29052    FrSe w-bnj15 29056    trClc-bnj18 29058
This theorem is referenced by:  bnj1137  29364  bnj1136  29366  bnj1175  29373  bnj1177  29375  bnj1413  29404  bnj1408  29405  bnj1417  29410  bnj1442  29418  bnj1452  29421
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057  df-bnj18 29059
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