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Theorem bnj1466 28753
Description: Technical lemma for bnj60 28762. 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 28524 . . . . . . 7  |-  ( w  e.  H  ->  A. f  w  e.  H )
54nfcii 2507 . . . . . 6  |-  F/_ f H
65nfuni 3956 . . . . 5  |-  F/_ f U. H
72, 6nfcxfr 2513 . . . 4  |-  F/_ f P
8 nfcv 2516 . . . . . 6  |-  F/_ f
x
9 nfcv 2516 . . . . . . 7  |-  F/_ f G
10 bnj1466.11 . . . . . . . 8  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
11 nfcv 2516 . . . . . . . . . 10  |-  F/_ f  pred ( x ,  A ,  R )
127, 11nfres 5081 . . . . . . . . 9  |-  F/_ f
( P  |`  pred (
x ,  A ,  R ) )
138, 12nfop 3935 . . . . . . . 8  |-  F/_ f <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
1410, 13nfcxfr 2513 . . . . . . 7  |-  F/_ f Z
159, 14nffv 5668 . . . . . 6  |-  F/_ f
( G `  Z
)
168, 15nfop 3935 . . . . 5  |-  F/_ f <. x ,  ( G `
 Z ) >.
1716nfsn 3802 . . . 4  |-  F/_ f { <. x ,  ( G `  Z )
>. }
187, 17nfun 3439 . . 3  |-  F/_ f
( P  u.  { <. x ,  ( G `
 Z ) >. } )
191, 18nfcxfr 2513 . 2  |-  F/_ f Q
2019nfcrii 2509 1  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
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 2366    =/= wne 2543   A.wral 2642   E.wrex 2643   {crab 2646   [.wsbc 3097    u. cun 3254    C_ wss 3256   (/)c0 3564   {csn 3750   <.cop 3753   U.cuni 3950   class class class wbr 4146   dom cdm 4811    |` cres 4813    Fn wfn 5382   ` cfv 5387    predc-bnj14 28383    FrSe w-bnj15 28387    trClc-bnj18 28389
This theorem is referenced by:  bnj1463  28755  bnj1491  28757
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 2361
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-res 4823  df-iota 5351  df-fv 5395
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