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Theorem bnj1467 29084
Description: Technical lemma for bnj60 29092. 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 2419 . . . . . . . . 9  |-  F/_ d  pred ( x ,  A ,  R )
5 bnj1467.8 . . . . . . . . . 10  |-  ( ta'  <->  [. y  /  x ]. ta )
6 nfcv 2419 . . . . . . . . . . 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 2599 . . . . . . . . . . . . . . . 16  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
109nfab 2423 . . . . . . . . . . . . . . 15  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
118, 10nfcxfr 2416 . . . . . . . . . . . . . 14  |-  F/_ d C
1211nfcri 2413 . . . . . . . . . . . . 13  |-  F/ d  f  e.  C
13 nfv 1605 . . . . . . . . . . . . 13  |-  F/ d dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
1412, 13nfan 1771 . . . . . . . . . . . 12  |-  F/ d ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
157, 14nfxfr 1557 . . . . . . . . . . 11  |-  F/ d ta
166, 15nfsbc 3012 . . . . . . . . . 10  |-  F/ d
[. y  /  x ]. ta
175, 16nfxfr 1557 . . . . . . . . 9  |-  F/ d ta'
184, 17nfrex 2598 . . . . . . . 8  |-  F/ d E. y  e.  pred  ( x ,  A ,  R ) ta'
1918nfab 2423 . . . . . . 7  |-  F/_ d { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
203, 19nfcxfr 2416 . . . . . 6  |-  F/_ d H
2120nfuni 3833 . . . . 5  |-  F/_ d U. H
222, 21nfcxfr 2416 . . . 4  |-  F/_ d P
23 nfcv 2419 . . . . . 6  |-  F/_ d
x
24 nfcv 2419 . . . . . . 7  |-  F/_ d G
25 bnj1467.11 . . . . . . . 8  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2622, 4nfres 4957 . . . . . . . . 9  |-  F/_ d
( P  |`  pred (
x ,  A ,  R ) )
2723, 26nfop 3812 . . . . . . . 8  |-  F/_ d <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2825, 27nfcxfr 2416 . . . . . . 7  |-  F/_ d Z
2924, 28nffv 5532 . . . . . 6  |-  F/_ d
( G `  Z
)
3023, 29nfop 3812 . . . . 5  |-  F/_ d <. x ,  ( G `
 Z ) >.
3130nfsn 3691 . . . 4  |-  F/_ d { <. x ,  ( G `  Z )
>. }
3222, 31nfun 3331 . . 3  |-  F/_ d
( P  u.  { <. x ,  ( G `
 Z ) >. } )
331, 32nfcxfr 2416 . 2  |-  F/_ d Q
3433nfcrii 2412 1  |-  ( w  e.  Q  ->  A. d  w  e.  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   [.wsbc 2991    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1463  29085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263
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