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Theorem bnj1446 33397
Description: Technical lemma for bnj60 33414. 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
bnj1446.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1446.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1446.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1446.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1446.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1446.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1446.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1446.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1446.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1446.10  |-  P  = 
U. H
bnj1446.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1446.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1446.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
Assertion
Ref Expression
bnj1446  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
Distinct variable groups:    A, d, x    B, f    G, d    R, d, x    f, d, x    y, d, x   
z, d
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, z, f, d)    ta( x, y, z, f, d)    A( y, z, f)    B( x, y, z, 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( y, z, f)    G( x, y, z, f)    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 bnj1446
StepHypRef Expression
1 bnj1446.12 . . . . 5  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
2 bnj1446.10 . . . . . . 7  |-  P  = 
U. H
3 bnj1446.9 . . . . . . . . 9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
4 nfcv 2629 . . . . . . . . . . 11  |-  F/_ d  pred ( x ,  A ,  R )
5 bnj1446.8 . . . . . . . . . . . 12  |-  ( ta'  <->  [. y  /  x ]. ta )
6 nfcv 2629 . . . . . . . . . . . . 13  |-  F/_ d
y
7 bnj1446.4 . . . . . . . . . . . . . 14  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
8 bnj1446.3 . . . . . . . . . . . . . . . . 17  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
9 nfre1 2925 . . . . . . . . . . . . . . . . . 18  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
109nfab 2633 . . . . . . . . . . . . . . . . 17  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
118, 10nfcxfr 2627 . . . . . . . . . . . . . . . 16  |-  F/_ d C
1211nfcri 2622 . . . . . . . . . . . . . . 15  |-  F/ d  f  e.  C
13 nfv 1683 . . . . . . . . . . . . . . 15  |-  F/ d dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
1412, 13nfan 1875 . . . . . . . . . . . . . 14  |-  F/ d ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
157, 14nfxfr 1625 . . . . . . . . . . . . 13  |-  F/ d ta
166, 15nfsbc 3353 . . . . . . . . . . . 12  |-  F/ d
[. y  /  x ]. ta
175, 16nfxfr 1625 . . . . . . . . . . 11  |-  F/ d ta'
184, 17nfrex 2927 . . . . . . . . . 10  |-  F/ d E. y  e.  pred  ( x ,  A ,  R ) ta'
1918nfab 2633 . . . . . . . . 9  |-  F/_ d { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
203, 19nfcxfr 2627 . . . . . . . 8  |-  F/_ d H
2120nfuni 4251 . . . . . . 7  |-  F/_ d U. H
222, 21nfcxfr 2627 . . . . . 6  |-  F/_ d P
23 nfcv 2629 . . . . . . . 8  |-  F/_ d
x
24 nfcv 2629 . . . . . . . . 9  |-  F/_ d G
25 bnj1446.11 . . . . . . . . . 10  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2622, 4nfres 5275 . . . . . . . . . . 11  |-  F/_ d
( P  |`  pred (
x ,  A ,  R ) )
2723, 26nfop 4229 . . . . . . . . . 10  |-  F/_ d <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2825, 27nfcxfr 2627 . . . . . . . . 9  |-  F/_ d Z
2924, 28nffv 5873 . . . . . . . 8  |-  F/_ d
( G `  Z
)
3023, 29nfop 4229 . . . . . . 7  |-  F/_ d <. x ,  ( G `
 Z ) >.
3130nfsn 4085 . . . . . 6  |-  F/_ d { <. x ,  ( G `  Z )
>. }
3222, 31nfun 3660 . . . . 5  |-  F/_ d
( P  u.  { <. x ,  ( G `
 Z ) >. } )
331, 32nfcxfr 2627 . . . 4  |-  F/_ d Q
34 nfcv 2629 . . . 4  |-  F/_ d
z
3533, 34nffv 5873 . . 3  |-  F/_ d
( Q `  z
)
36 bnj1446.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
37 nfcv 2629 . . . . . . 7  |-  F/_ d  pred ( z ,  A ,  R )
3833, 37nfres 5275 . . . . . 6  |-  F/_ d
( Q  |`  pred (
z ,  A ,  R ) )
3934, 38nfop 4229 . . . . 5  |-  F/_ d <. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
4036, 39nfcxfr 2627 . . . 4  |-  F/_ d W
4124, 40nffv 5873 . . 3  |-  F/_ d
( G `  W
)
4235, 41nfeq 2640 . 2  |-  F/ d ( Q `  z
)  =  ( G `
 W )
4342nfri 1822 1  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   [.wsbc 3331    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588    predc-bnj14 33037    FrSe w-bnj15 33041    trClc-bnj18 33043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011  df-iota 5551  df-fv 5596
This theorem is referenced by:  bnj1450  33402  bnj1463  33407
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