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Theorem bnj1374 29848
Description: Technical lemma for bnj60 29879. 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 29659 . . . . 5  |-  ( f  e.  H  ->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3 rexex 2879 . . . . 5  |-  ( E. y  e.  pred  (
x ,  A ,  R ) ta'  ->  E. y ta' )
42, 3syl 17 . . . 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 29847 . . . . 5  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
1110exbii 1712 . . . 4  |-  ( E. y ta'  <->  E. y ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
124, 11sylib 199 . . 3  |-  ( f  e.  H  ->  E. y
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
13 exsimpl 1723 . . 3  |-  ( E. y ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) )  ->  E. y  f  e.  C )
1412, 13syl 17 . 2  |-  ( f  e.  H  ->  E. y 
f  e.  C )
1514bnj937 29591 1  |-  ( f  e.  H  ->  f  e.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775   [.wsbc 3299    u. cun 3434    C_ wss 3436   (/)c0 3761   {csn 3998   <.cop 4004   class class class wbr 4423   dom cdm 4853    |` cres 4855    Fn wfn 5596   ` cfv 5601    predc-bnj14 29501    FrSe w-bnj15 29505    trClc-bnj18 29507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-iun 4301  df-br 4424  df-bnj14 29502  df-bnj18 29508
This theorem is referenced by:  bnj1384  29849
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