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Theorem bnj1467 32358
Description: Technical lemma for bnj60 32366. 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
bnj1467.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1467.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1467.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1467.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1467.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1467.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1467.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1467.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1467.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1467.10  |-  P  = 
U. H
bnj1467.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1467.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
Assertion
Ref Expression
bnj1467  |-  ( w  e.  Q  ->  A. d  w  e.  Q )
Distinct variable groups:    A, d, w, x    B, f    w, C    G, d, w    w, H    w, P    R, d, w, x    w, Z    f,
d, w, x    y,
d, x
Allowed substitution hints:    ps( x, y, w, f, d)    ch( x, y, w, f, d)    ta( x, y, w, f, d)    A( y, f)    B( x, y, w, d)    C( x, y, f, d)    D( x, y, w, f, d)    P( x, y, f, d)    Q( x, y, w, f, d)    R( y, f)    G( x, y, f)    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 bnj1467
StepHypRef Expression
1 bnj1467.12 . . 3  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
2 bnj1467.10 . . . . 5  |-  P  = 
U. H
3 bnj1467.9 . . . . . . 7  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
4 nfcv 2614 . . . . . . . . 9  |-  F/_ d  pred ( x ,  A ,  R )
5 bnj1467.8 . . . . . . . . . 10  |-  ( ta'  <->  [. y  /  x ]. ta )
6 nfcv 2614 . . . . . . . . . . 11  |-  F/_ d
y
7 bnj1467.4 . . . . . . . . . . . 12  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
8 bnj1467.3 . . . . . . . . . . . . . . 15  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
9 nfre1 2885 . . . . . . . . . . . . . . . 16  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
109nfab 2618 . . . . . . . . . . . . . . 15  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
118, 10nfcxfr 2612 . . . . . . . . . . . . . 14  |-  F/_ d C
1211nfcri 2607 . . . . . . . . . . . . 13  |-  F/ d  f  e.  C
13 nfv 1674 . . . . . . . . . . . . 13  |-  F/ d dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
1412, 13nfan 1865 . . . . . . . . . . . 12  |-  F/ d ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
157, 14nfxfr 1616 . . . . . . . . . . 11  |-  F/ d ta
166, 15nfsbc 3310 . . . . . . . . . 10  |-  F/ d
[. y  /  x ]. ta
175, 16nfxfr 1616 . . . . . . . . 9  |-  F/ d ta'
184, 17nfrex 2884 . . . . . . . 8  |-  F/ d E. y  e.  pred  ( x ,  A ,  R ) ta'
1918nfab 2618 . . . . . . 7  |-  F/_ d { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
203, 19nfcxfr 2612 . . . . . 6  |-  F/_ d H
2120nfuni 4200 . . . . 5  |-  F/_ d U. H
222, 21nfcxfr 2612 . . . 4  |-  F/_ d P
23 nfcv 2614 . . . . . 6  |-  F/_ d
x
24 nfcv 2614 . . . . . . 7  |-  F/_ d G
25 bnj1467.11 . . . . . . . 8  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2622, 4nfres 5215 . . . . . . . . 9  |-  F/_ d
( P  |`  pred (
x ,  A ,  R ) )
2723, 26nfop 4178 . . . . . . . 8  |-  F/_ d <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2825, 27nfcxfr 2612 . . . . . . 7  |-  F/_ d Z
2924, 28nffv 5801 . . . . . 6  |-  F/_ d
( G `  Z
)
3023, 29nfop 4178 . . . . 5  |-  F/_ d <. x ,  ( G `
 Z ) >.
3130nfsn 4037 . . . 4  |-  F/_ d { <. x ,  ( G `  Z )
>. }
3222, 31nfun 3615 . . 3  |-  F/_ d
( P  u.  { <. x ,  ( G `
 Z ) >. } )
331, 32nfcxfr 2612 . 2  |-  F/_ d Q
3433nfcrii 2606 1  |-  ( w  e.  Q  ->  A. d  w  e.  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2437    =/= wne 2645   A.wral 2796   E.wrex 2797   {crab 2800   [.wsbc 3288    u. cun 3429    C_ wss 3431   (/)c0 3740   {csn 3980   <.cop 3986   U.cuni 4194   class class class wbr 4395   dom cdm 4943    |` cres 4945    Fn wfn 5516   ` cfv 5521    predc-bnj14 31989    FrSe w-bnj15 31993    trClc-bnj18 31995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-xp 4949  df-res 4955  df-iota 5484  df-fv 5529
This theorem is referenced by:  bnj1463  32359
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