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Theorem bnj1444 29073
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
bnj1444.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1444.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1444.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1444.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1444.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1444.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1444.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1444.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1444.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1444.10  |-  P  = 
U. H
bnj1444.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1444.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1444.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1444.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1444.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1444.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1444.17  |-  ( th  <->  ( ch  /\  z  e.  E ) )
bnj1444.18  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
bnj1444.19  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
bnj1444.20  |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e. 
dom  f ) )
Assertion
Ref Expression
bnj1444  |-  ( rh 
->  A. y rh )
Distinct variable groups:    y, A    y, D    y, E    y, R    y, f    ps, y    x, y    y, z
Allowed substitution hints:    ps( x, z, f, d)    ch( x, y, z, f, d)    th( x, y, z, f, d)    ta( x, y, z, f, d)    et( x, y, z, f, d)    ze( x, y, z, f, d)    rh( x, y, z, f, d)    A( x, z, f, d)    B( x, y, z, f, d)    C( x, y, z, f, d)    D( x, z, f, d)    P( x, y, z, f, d)    Q( x, y, z, f, d)    R( x, z, f, d)    E( x, z, f, d)    G( x, y, z, f, d)    H( x, y, z, f, d)    W( x, y, z, f, d)    Y( x, y, z, f, d)    Z( x, y, z, f, d)    ta'( x, y, z, f, d)

Proof of Theorem bnj1444
StepHypRef Expression
1 bnj1444.20 . . 3  |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e. 
dom  f ) )
2 bnj1444.19 . . . . 5  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
3 bnj1444.17 . . . . . . 7  |-  ( th  <->  ( ch  /\  z  e.  E ) )
4 bnj1444.7 . . . . . . . . 9  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
5 nfv 1605 . . . . . . . . . 10  |-  F/ y ps
6 nfv 1605 . . . . . . . . . 10  |-  F/ y  x  e.  D
7 nfra1 2593 . . . . . . . . . 10  |-  F/ y A. y  e.  D  -.  y R x
85, 6, 7nf3an 1774 . . . . . . . . 9  |-  F/ y ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
94, 8nfxfr 1557 . . . . . . . 8  |-  F/ y ch
10 nfv 1605 . . . . . . . 8  |-  F/ y  z  e.  E
119, 10nfan 1771 . . . . . . 7  |-  F/ y ( ch  /\  z  e.  E )
123, 11nfxfr 1557 . . . . . 6  |-  F/ y th
13 nfv 1605 . . . . . 6  |-  F/ y  z  e.  trCl (
x ,  A ,  R )
1412, 13nfan 1771 . . . . 5  |-  F/ y ( th  /\  z  e.  trCl ( x ,  A ,  R ) )
152, 14nfxfr 1557 . . . 4  |-  F/ y ze
16 bnj1444.9 . . . . . 6  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
17 nfre1 2599 . . . . . . 7  |-  F/ y E. y  e.  pred  ( x ,  A ,  R ) ta'
1817nfab 2423 . . . . . 6  |-  F/_ y { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
1916, 18nfcxfr 2416 . . . . 5  |-  F/_ y H
2019nfcri 2413 . . . 4  |-  F/ y  f  e.  H
21 nfv 1605 . . . 4  |-  F/ y  z  e.  dom  f
2215, 20, 21nf3an 1774 . . 3  |-  F/ y ( ze  /\  f  e.  H  /\  z  e.  dom  f )
231, 22nfxfr 1557 . 2  |-  F/ y rh
2423nfri 1742 1  |-  ( rh 
->  A. y rh )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   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   U.cuni 3827   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:  bnj1450  29080
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
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