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Theorem bnj906 29278
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 6653 . . . . . . . 8  |-  1o  e.  om
2 1n0 6510 . . . . . . . 8  |-  1o  =/=  (/)
3 eldifsn 3762 . . . . . . . 8  |-  ( 1o  e.  ( om  \  { (/)
} )  <->  ( 1o  e.  om  /\  1o  =/=  (/) ) )
41, 2, 3mpbir2an 886 . . . . . . 7  |-  1o  e.  ( om  \  { (/) } )
5 ne0i 3474 . . . . . . 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 2296 . . . . . . 7  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
107, 8, 9bnj852 29269 . . . . . 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 3556 . . . . . 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 2179 . . . . 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 29061 . . . 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 2820 . . . 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 2284 . . 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 29144 . 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 2663 . . . . . 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 2562 . . . . . 6  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
) )
23 19.41v 1854 . . . . . . 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 2296 . . . . . . 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 29277 . . . . . 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 5541 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
3029ssiun2s 3962 . . . . . 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 3961 . . . . . 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 29274 . . . . . 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 3238 . . . . 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 3202 . . . 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 3226 . . 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 1624 . 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 1531    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   U_ciun 3921   suc csuc 4410   omcom 4672   dom cdm 4705    Fn wfn 5266   ` cfv 5271   1oc1o 6488    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1137  29341  bnj1136  29343  bnj1175  29350  bnj1177  29352  bnj1413  29381  bnj1408  29382  bnj1417  29387  bnj1442  29395  bnj1452  29398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036
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