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Theorem bnj1444 29486
Description: Technical lemma for bnj60 29505. 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 1630 . . . . . . . . . 10  |-  F/ y ps
6 nfv 1630 . . . . . . . . . 10  |-  F/ y  x  e.  D
7 nfra1 2758 . . . . . . . . . 10  |-  F/ y A. y  e.  D  -.  y R x
85, 6, 7nf3an 1850 . . . . . . . . 9  |-  F/ y ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
94, 8nfxfr 1580 . . . . . . . 8  |-  F/ y ch
10 nfv 1630 . . . . . . . 8  |-  F/ y  z  e.  E
119, 10nfan 1847 . . . . . . 7  |-  F/ y ( ch  /\  z  e.  E )
123, 11nfxfr 1580 . . . . . 6  |-  F/ y th
13 nfv 1630 . . . . . 6  |-  F/ y  z  e.  trCl (
x ,  A ,  R )
1412, 13nfan 1847 . . . . 5  |-  F/ y ( th  /\  z  e.  trCl ( x ,  A ,  R ) )
152, 14nfxfr 1580 . . . 4  |-  F/ y ze
16 bnj1444.9 . . . . . 6  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
17 nfre1 2764 . . . . . . 7  |-  F/ y E. y  e.  pred  ( x ,  A ,  R ) ta'
1817nfab 2578 . . . . . 6  |-  F/_ y { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
1916, 18nfcxfr 2571 . . . . 5  |-  F/_ y H
2019nfcri 2568 . . . 4  |-  F/ y  f  e.  H
21 nfv 1630 . . . 4  |-  F/ y  z  e.  dom  f
2215, 20, 21nf3an 1850 . . 3  |-  F/ y ( ze  /\  f  e.  H  /\  z  e.  dom  f )
231, 22nfxfr 1580 . 2  |-  F/ y rh
2423nfri 1779 1  |-  ( rh 
->  A. y rh )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   [.wsbc 3163    u. cun 3320    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U.cuni 4017   class class class wbr 4215   dom cdm 4881    |` cres 4883    Fn wfn 5452   ` cfv 5457    predc-bnj14 29126    FrSe w-bnj15 29130    trClc-bnj18 29132
This theorem is referenced by:  bnj1450  29493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713
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