Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj906 Structured version   Unicode version

Theorem bnj906 33067
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 7285 . . . . . . . 8  |-  1o  e.  om
2 1n0 7142 . . . . . . . 8  |-  1o  =/=  (/)
3 eldifsn 4152 . . . . . . . 8  |-  ( 1o  e.  ( om  \  { (/)
} )  <->  ( 1o  e.  om  /\  1o  =/=  (/) ) )
41, 2, 3mpbir2an 918 . . . . . . 7  |-  1o  e.  ( om  \  { (/) } )
5 ne0i 3791 . . . . . . 7  |-  ( 1o  e.  ( om  \  { (/)
} )  ->  ( om  \  { (/) } )  =/=  (/) )
64, 5ax-mp 5 . . . . . 6  |-  ( om 
\  { (/) } )  =/=  (/)
7 biid 236 . . . . . . 7  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
8 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 ) ) )
9 eqid 2467 . . . . . . 7  |-  ( om 
\  { (/) } )  =  ( om  \  { (/)
} )
107, 8, 9bnj852 33058 . . . . . 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 3917 . . . . . 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 663 . . . . 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 2303 . . . . 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 32852 . . . 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 3133 . . . 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 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 ) ) ) )
17 abid 2454 . . 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 32933 . 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 997 . . . . . . 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 2932 . . . . . 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 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 ) )
22 df-rex 2820 . . . . . 6  |-  ( E. n  e.  ( om 
\  { (/) } ) ( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
) )
23 19.41v 1945 . . . . . . 7  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  <->  ( E. n  n  e.  ( om  \  { (/) } )  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
2423simprbi 464 . . . . . 6  |-  ( E. n ( n  e.  ( om  \  { (/)
} )  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  (/) )  =  pred ( X ,  A ,  R ) )
2522, 24sylbi 195 . . . . 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 2467 . . . . . . 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 33066 . . . . . 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 5864 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
3029ssiun2s 4369 . . . . . 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 4368 . . . . . 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 33063 . . . . . 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 3551 . . . . 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 3514 . . . 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 3539 . . 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 1698 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   U_ciun 4325   suc csuc 4880   dom cdm 4999    Fn wfn 5581   ` cfv 5586   omcom 6678   1oc1o 7120    predc-bnj14 32820    FrSe w-bnj15 32824    trClc-bnj18 32826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-reg 8014  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-1o 7127  df-bnj17 32819  df-bnj14 32821  df-bnj13 32823  df-bnj15 32825  df-bnj18 32827
This theorem is referenced by:  bnj1137  33130  bnj1136  33132  bnj1175  33139  bnj1177  33141  bnj1413  33170  bnj1408  33171  bnj1417  33176  bnj1442  33184  bnj1452  33187
  Copyright terms: Public domain W3C validator