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Theorem bnj1466 33589
Description: Technical lemma for bnj60 33598. 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
bnj1466.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1466.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1466.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1466.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1466.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1466.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1466.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1466.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1466.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1466.10  |-  P  = 
U. H
bnj1466.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1466.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
Assertion
Ref Expression
bnj1466  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
Distinct variable groups:    A, f, w    f, G, w    w, H    w, P    R, f, w    w, Z    x, f, w
Allowed substitution hints:    ps( x, y, w, f, d)    ch( x, y, w, f, d)    ta( x, y, w, f, d)    A( x, y, d)    B( x, y, w, f, d)    C( x, y, w, f, d)    D( x, y, w, f, d)    P( x, y, f, d)    Q( x, y, w, f, d)    R( x, y, d)    G( x, y, d)    H( x, y, f, d)    Y( x, y, w, f, d)    Z( x, y, f, d)    ta'( x, y, w, f, d)

Proof of Theorem bnj1466
StepHypRef Expression
1 bnj1466.12 . . 3  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
2 bnj1466.10 . . . . 5  |-  P  = 
U. H
3 bnj1466.9 . . . . . . . 8  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
43bnj1317 33360 . . . . . . 7  |-  ( w  e.  H  ->  A. f  w  e.  H )
54nfcii 2619 . . . . . 6  |-  F/_ f H
65nfuni 4257 . . . . 5  |-  F/_ f U. H
72, 6nfcxfr 2627 . . . 4  |-  F/_ f P
8 nfcv 2629 . . . . . 6  |-  F/_ f
x
9 nfcv 2629 . . . . . . 7  |-  F/_ f G
10 bnj1466.11 . . . . . . . 8  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
11 nfcv 2629 . . . . . . . . . 10  |-  F/_ f  pred ( x ,  A ,  R )
127, 11nfres 5281 . . . . . . . . 9  |-  F/_ f
( P  |`  pred (
x ,  A ,  R ) )
138, 12nfop 4235 . . . . . . . 8  |-  F/_ f <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
1410, 13nfcxfr 2627 . . . . . . 7  |-  F/_ f Z
159, 14nffv 5879 . . . . . 6  |-  F/_ f
( G `  Z
)
168, 15nfop 4235 . . . . 5  |-  F/_ f <. x ,  ( G `
 Z ) >.
1716nfsn 4091 . . . 4  |-  F/_ f { <. x ,  ( G `  Z )
>. }
187, 17nfun 3665 . . 3  |-  F/_ f
( P  u.  { <. x ,  ( G `
 Z ) >. } )
191, 18nfcxfr 2627 . 2  |-  F/_ f Q
2019nfcrii 2621 1  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   [.wsbc 3336    u. cun 3479    C_ wss 3481   (/)c0 3790   {csn 4033   <.cop 4039   U.cuni 4251   class class class wbr 4453   dom cdm 5005    |` cres 5007    Fn wfn 5589   ` cfv 5594    predc-bnj14 33221    FrSe w-bnj15 33225    trClc-bnj18 33227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-res 5017  df-iota 5557  df-fv 5602
This theorem is referenced by:  bnj1463  33591  bnj1491  33593
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