Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1374 Unicode version

Theorem bnj1374 29061
Description: Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1374.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1374.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1374.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1374.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1374.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1374.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1374.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1374.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1374.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
Assertion
Ref Expression
bnj1374  |-  ( f  e.  H  ->  f  e.  C )
Distinct variable groups:    x, A    B, f    y, C    x, R    f, d, x    y,
f, x
Allowed substitution hints:    ps( x, y, f, d)    ch( x, y, f, d)    ta( x, y, f, d)    A( y, f, d)    B( x, y, d)    C( x, f, d)    D( x, y, f, d)    R( y, f, d)    G( x, y, f, d)    H( x, y, f, d)    Y( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1374
StepHypRef Expression
1 bnj1374.9 . . . . . 6  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
21bnj1436 28872 . . . . 5  |-  ( f  e.  H  ->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3 rexex 2602 . . . . 5  |-  ( E. y  e.  pred  (
x ,  A ,  R ) ta'  ->  E. y ta' )
42, 3syl 15 . . . 4  |-  ( f  e.  H  ->  E. y ta' )
5 bnj1374.1 . . . . . 6  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
6 bnj1374.2 . . . . . 6  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
7 bnj1374.3 . . . . . 6  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
8 bnj1374.4 . . . . . 6  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
9 bnj1374.8 . . . . . 6  |-  ( ta'  <->  [. y  /  x ]. ta )
105, 6, 7, 8, 9bnj1373 29060 . . . . 5  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
1110exbii 1569 . . . 4  |-  ( E. y ta'  <->  E. y ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
124, 11sylib 188 . . 3  |-  ( f  e.  H  ->  E. y
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
13 exsimpl 1579 . . 3  |-  ( E. y ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) )  ->  E. y  f  e.  C )
1412, 13syl 15 . 2  |-  ( f  e.  H  ->  E. y 
f  e.  C )
1514bnj937 28803 1  |-  ( f  e.  H  ->  f  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   [.wsbc 2991    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1384  29062
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-br 4024  df-bnj14 28714  df-bnj18 28720
  Copyright terms: Public domain W3C validator