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Theorem bnj1449 28757
Description: Technical lemma for bnj60 28771. 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
bnj1449.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1449.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1449.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1449.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1449.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1449.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1449.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1449.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1449.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1449.10  |-  P  = 
U. H
bnj1449.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1449.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1449.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1449.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1449.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1449.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1449.17  |-  ( th  <->  ( ch  /\  z  e.  E ) )
bnj1449.18  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
bnj1449.19  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
Assertion
Ref Expression
bnj1449  |-  ( ze 
->  A. f ze )
Distinct variable groups:    A, f    f, E    R, f    x, f   
y, f    z, f
Allowed substitution hints:    ps( x, y, 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)    A( x, y, z, d)    B( x, y, z, f, d)    C( x, y, z, f, d)    D( x, y, z, f, d)    P( x, y, z, f, d)    Q( x, y, z, f, d)    R( x, y, z, d)    E( x, y, z, 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 bnj1449
StepHypRef Expression
1 bnj1449.19 . . 3  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
2 bnj1449.17 . . . . 5  |-  ( th  <->  ( ch  /\  z  e.  E ) )
3 bnj1449.7 . . . . . . 7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
4 bnj1449.6 . . . . . . . . 9  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
5 nfv 1626 . . . . . . . . . 10  |-  F/ f  R  FrSe  A
6 bnj1449.5 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  -.  E. f ta }
7 nfe1 1739 . . . . . . . . . . . . . 14  |-  F/ f E. f ta
87nfn 1801 . . . . . . . . . . . . 13  |-  F/ f  -.  E. f ta
9 nfcv 2525 . . . . . . . . . . . . 13  |-  F/_ f A
108, 9nfrab 2834 . . . . . . . . . . . 12  |-  F/_ f { x  e.  A  |  -.  E. f ta }
116, 10nfcxfr 2522 . . . . . . . . . . 11  |-  F/_ f D
12 nfcv 2525 . . . . . . . . . . 11  |-  F/_ f (/)
1311, 12nfne 2643 . . . . . . . . . 10  |-  F/ f  D  =/=  (/)
145, 13nfan 1836 . . . . . . . . 9  |-  F/ f ( R  FrSe  A  /\  D  =/=  (/) )
154, 14nfxfr 1576 . . . . . . . 8  |-  F/ f ps
1611nfcri 2519 . . . . . . . 8  |-  F/ f  x  e.  D
17 nfv 1626 . . . . . . . . 9  |-  F/ f  -.  y R x
1811, 17nfral 2704 . . . . . . . 8  |-  F/ f A. y  e.  D  -.  y R x
1915, 16, 18nf3an 1839 . . . . . . 7  |-  F/ f ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
203, 19nfxfr 1576 . . . . . 6  |-  F/ f ch
21 nfv 1626 . . . . . 6  |-  F/ f  z  e.  E
2220, 21nfan 1836 . . . . 5  |-  F/ f ( ch  /\  z  e.  E )
232, 22nfxfr 1576 . . . 4  |-  F/ f th
24 nfv 1626 . . . 4  |-  F/ f  z  e.  trCl (
x ,  A ,  R )
2523, 24nfan 1836 . . 3  |-  F/ f ( th  /\  z  e.  trCl ( x ,  A ,  R ) )
261, 25nfxfr 1576 . 2  |-  F/ f ze
2726nfri 1770 1  |-  ( ze 
->  A. f ze )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2375    =/= wne 2552   A.wral 2651   E.wrex 2652   {crab 2655   [.wsbc 3106    u. cun 3263    C_ wss 3265   (/)c0 3573   {csn 3759   <.cop 3762   U.cuni 3959   class class class wbr 4155   dom cdm 4820    |` cres 4822    Fn wfn 5391   ` cfv 5396    predc-bnj14 28392    FrSe w-bnj15 28396    trClc-bnj18 28398
This theorem is referenced by:  bnj1450  28759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rab 2660
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