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Theorem bnj1448 32035
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
bnj1448.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1448.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1448.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1448.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1448.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1448.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1448.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1448.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1448.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1448.10  |-  P  = 
U. H
bnj1448.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1448.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1448.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
Assertion
Ref Expression
bnj1448  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. f
( Q `  z
)  =  ( G `
 W ) )
Distinct variable groups:    A, f    f, G    R, f    x, f   
z, f
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, z, f, d)    ta( x, y, z, f, d)    A( x, y, z, 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, y, z, d)    G( x, y, z, 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 bnj1448
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1448.12 . . . . 5  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
2 bnj1448.10 . . . . . . 7  |-  P  = 
U. H
3 bnj1448.9 . . . . . . . . . 10  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
43bnj1317 31812 . . . . . . . . 9  |-  ( w  e.  H  ->  A. f  w  e.  H )
54nfcii 2568 . . . . . . . 8  |-  F/_ f H
65nfuni 4095 . . . . . . 7  |-  F/_ f U. H
72, 6nfcxfr 2574 . . . . . 6  |-  F/_ f P
8 nfcv 2577 . . . . . . . 8  |-  F/_ f
x
9 nfcv 2577 . . . . . . . . 9  |-  F/_ f G
10 bnj1448.11 . . . . . . . . . 10  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
11 nfcv 2577 . . . . . . . . . . . 12  |-  F/_ f  pred ( x ,  A ,  R )
127, 11nfres 5110 . . . . . . . . . . 11  |-  F/_ f
( P  |`  pred (
x ,  A ,  R ) )
138, 12nfop 4073 . . . . . . . . . 10  |-  F/_ f <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
1410, 13nfcxfr 2574 . . . . . . . . 9  |-  F/_ f Z
159, 14nffv 5696 . . . . . . . 8  |-  F/_ f
( G `  Z
)
168, 15nfop 4073 . . . . . . 7  |-  F/_ f <. x ,  ( G `
 Z ) >.
1716nfsn 3932 . . . . . 6  |-  F/_ f { <. x ,  ( G `  Z )
>. }
187, 17nfun 3510 . . . . 5  |-  F/_ f
( P  u.  { <. x ,  ( G `
 Z ) >. } )
191, 18nfcxfr 2574 . . . 4  |-  F/_ f Q
20 nfcv 2577 . . . 4  |-  F/_ f
z
2119, 20nffv 5696 . . 3  |-  F/_ f
( Q `  z
)
22 bnj1448.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
23 nfcv 2577 . . . . . . 7  |-  F/_ f  pred ( z ,  A ,  R )
2419, 23nfres 5110 . . . . . 6  |-  F/_ f
( Q  |`  pred (
z ,  A ,  R ) )
2520, 24nfop 4073 . . . . 5  |-  F/_ f <. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
2622, 25nfcxfr 2574 . . . 4  |-  F/_ f W
279, 26nffv 5696 . . 3  |-  F/_ f
( G `  W
)
2821, 27nfeq 2584 . 2  |-  F/ f ( Q `  z
)  =  ( G `
 W )
2928nfri 1808 1  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. f
( 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  bnj1463  32043
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