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Theorem bnj906 34408
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 7280 . . . . . . . 8  |-  1o  e.  om
2 1n0 7137 . . . . . . . 8  |-  1o  =/=  (/)
3 eldifsn 4141 . . . . . . . 8  |-  ( 1o  e.  ( om  \  { (/)
} )  <->  ( 1o  e.  om  /\  1o  =/=  (/) ) )
41, 2, 3mpbir2an 918 . . . . . . 7  |-  1o  e.  ( om  \  { (/) } )
54ne0ii 3790 . . . . . 6  |-  ( om 
\  { (/) } )  =/=  (/)
6 biid 236 . . . . . . 7  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7 biid 236 . . . . . . 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 ) ) )
8 eqid 2454 . . . . . . 7  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
96, 7, 8bnj852 34399 . . . . . 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 ) ) ) )
10 r19.2z 3906 . . . . . 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 ) ) ) )
115, 9, 10sylancr 661 . . . . 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 ) ) ) )
12 euex 2310 . . . . 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 ) ) ) )
1311, 12bnj31 34192 . . . 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 ) ) ) )
14 rexcom4 3126 . . . 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 ) ) ) )
1513, 14sylib 196 . . 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 ) ) ) )
16 abid 2441 . . 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 ) ) ) )
1715, 16bnj1198 34274 . 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 ) ) ) } )
18 simp2 995 . . . . . . 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 ) )
1918reximi 2922 . . . . . 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 ) )
2016, 19sylbi 195 . . . . 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 ) )
21 df-rex 2810 . . . . . 6  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
) )
22 19.41v 1776 . . . . . . 7  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  <->  ( E. n  n  e.  ( om  \  { (/) } )  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
2322simprbi 462 . . . . . 6  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  (/) )  =  pred ( X ,  A ,  R ) )
2421, 23sylbi 195 . . . . 5  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2520, 24syl 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 ) )
26 eqid 2454 . . . . . . 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 ) ) ) }
278, 26bnj900 34407 . . . . . 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 )
28 fveq2 5848 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
2928ssiun2s 4359 . . . . . 6  |-  ( (/)  e.  dom  f  ->  (
f `  (/) )  C_  U_ i  e.  dom  f
( f `  i
) )
3027, 29syl 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 ) )
31 ssiun2 4358 . . . . . 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 ) )
326, 7, 8, 26bnj882 34404 . . . . . 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 )
3331, 32syl6sseqr 3536 . . . . 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 ) )
3430, 33sstrd 3499 . . . 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 ) )
3525, 34eqsstr3d 3524 . . 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
) )
3635exlimiv 1727 . 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
) )
3717, 36syl 16 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   E!weu 2284   {cab 2439    =/= wne 2649   A.wral 2804   E.wrex 2805    \ cdif 3458    C_ wss 3461   (/)c0 3783   {csn 4016   U_ciun 4315   suc csuc 4869   dom cdm 4988    Fn wfn 5565   ` cfv 5570   omcom 6673   1oc1o 7115    predc-bnj14 34160    FrSe w-bnj15 34164    trClc-bnj18 34166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-bnj17 34159  df-bnj14 34161  df-bnj13 34163  df-bnj15 34165  df-bnj18 34167
This theorem is referenced by:  bnj1137  34471  bnj1136  34473  bnj1175  34480  bnj1177  34482  bnj1413  34511  bnj1408  34512  bnj1417  34517  bnj1442  34525  bnj1452  34528
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