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Theorem bnj1447 32034
Description: Technical lemma for bnj60 32050. 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
bnj1447.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1447.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1447.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1447.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1447.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1447.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1447.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1447.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1447.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1447.10  |-  P  = 
U. H
bnj1447.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1447.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1447.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
Assertion
Ref Expression
bnj1447  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. y
( Q `  z
)  =  ( G `
 W ) )
Distinct variable groups:    y, A    y, G    y, R    x, y    y, z
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, z, f, d)    ta( x, y, z, f, d)    A( x, z, f, d)    B( x, y, z, f, d)    C( x, y, z, f, d)    D( x, y, z, f, d)    P( x, y, z, f, d)    Q( x, y, z, f, d)    R( x, z, f, d)    G( x, 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 bnj1447
StepHypRef Expression
1 bnj1447.12 . . . . 5  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
2 bnj1447.10 . . . . . . 7  |-  P  = 
U. H
3 bnj1447.9 . . . . . . . . 9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
4 nfre1 2770 . . . . . . . . . 10  |-  F/ y E. y  e.  pred  ( x ,  A ,  R ) ta'
54nfab 2581 . . . . . . . . 9  |-  F/_ y { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
63, 5nfcxfr 2574 . . . . . . . 8  |-  F/_ y H
76nfuni 4095 . . . . . . 7  |-  F/_ y U. H
82, 7nfcxfr 2574 . . . . . 6  |-  F/_ y P
9 nfcv 2577 . . . . . . . 8  |-  F/_ y
x
10 nfcv 2577 . . . . . . . . 9  |-  F/_ y G
11 bnj1447.11 . . . . . . . . . 10  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
12 nfcv 2577 . . . . . . . . . . . 12  |-  F/_ y  pred ( x ,  A ,  R )
138, 12nfres 5110 . . . . . . . . . . 11  |-  F/_ y
( P  |`  pred (
x ,  A ,  R ) )
149, 13nfop 4073 . . . . . . . . . 10  |-  F/_ y <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
1511, 14nfcxfr 2574 . . . . . . . . 9  |-  F/_ y Z
1610, 15nffv 5696 . . . . . . . 8  |-  F/_ y
( G `  Z
)
179, 16nfop 4073 . . . . . . 7  |-  F/_ y <. x ,  ( G `
 Z ) >.
1817nfsn 3932 . . . . . 6  |-  F/_ y { <. x ,  ( G `  Z )
>. }
198, 18nfun 3510 . . . . 5  |-  F/_ y
( P  u.  { <. x ,  ( G `
 Z ) >. } )
201, 19nfcxfr 2574 . . . 4  |-  F/_ y Q
21 nfcv 2577 . . . 4  |-  F/_ y
z
2220, 21nffv 5696 . . 3  |-  F/_ y
( Q `  z
)
23 bnj1447.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
24 nfcv 2577 . . . . . . 7  |-  F/_ y  pred ( z ,  A ,  R )
2520, 24nfres 5110 . . . . . 6  |-  F/_ y
( Q  |`  pred (
z ,  A ,  R ) )
2621, 25nfop 4073 . . . . 5  |-  F/_ y <. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
2723, 26nfcxfr 2574 . . . 4  |-  F/_ y W
2810, 27nffv 5696 . . 3  |-  F/_ y
( G `  W
)
2922, 28nfeq 2584 . 2  |-  F/ y ( Q `  z
)  =  ( G `
 W )
3029nfri 1808 1  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. y
( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2427    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   [.wsbc 3184    u. cun 3324    C_ wss 3326   (/)c0 3635   {csn 3875   <.cop 3881   U.cuni 4089   class class class wbr 4290   dom cdm 4838    |` cres 4840    Fn wfn 5411   ` cfv 5416    predc-bnj14 31673    FrSe w-bnj15 31677    trClc-bnj18 31679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-xp 4844  df-res 4850  df-iota 5379  df-fv 5424
This theorem is referenced by:  bnj1450  32038
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