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Theorem bnj1374 29912
Description: Technical lemma for bnj60 29943. 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 29723 . . . . 5  |-  ( f  e.  H  ->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3 rexex 2843 . . . . 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 29911 . . . . 5  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
1110exbii 1726 . . . 4  |-  ( E. y ta'  <->  E. y ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
124, 11sylib 201 . . 3  |-  ( f  e.  H  ->  E. y
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
13 exsimpl 1737 . . 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 29655 1  |-  ( f  e.  H  ->  f  e.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   [.wsbc 3255    u. cun 3388    C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   class class class wbr 4395   dom cdm 4839    |` cres 4841    Fn wfn 5584   ` cfv 5589    predc-bnj14 29565    FrSe w-bnj15 29569    trClc-bnj18 29571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-br 4396  df-bnj14 29566  df-bnj18 29572
This theorem is referenced by:  bnj1384  29913
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