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Theorem bnj1446 29351
Description: Technical lemma for bnj60 29368. 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 2571 . . . . . . . . . . 11  |-  F/_ d  pred ( x ,  A ,  R )
5 bnj1446.8 . . . . . . . . . . . 12  |-  ( ta'  <->  [. y  /  x ]. ta )
6 nfcv 2571 . . . . . . . . . . . . 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 2754 . . . . . . . . . . . . . . . . . 18  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
109nfab 2575 . . . . . . . . . . . . . . . . 17  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
118, 10nfcxfr 2568 . . . . . . . . . . . . . . . 16  |-  F/_ d C
1211nfcri 2565 . . . . . . . . . . . . . . 15  |-  F/ d  f  e.  C
13 nfv 1629 . . . . . . . . . . . . . . 15  |-  F/ d dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
1412, 13nfan 1846 . . . . . . . . . . . . . 14  |-  F/ d ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
157, 14nfxfr 1579 . . . . . . . . . . . . 13  |-  F/ d ta
166, 15nfsbc 3174 . . . . . . . . . . . 12  |-  F/ d
[. y  /  x ]. ta
175, 16nfxfr 1579 . . . . . . . . . . 11  |-  F/ d ta'
184, 17nfrex 2753 . . . . . . . . . 10  |-  F/ d E. y  e.  pred  ( x ,  A ,  R ) ta'
1918nfab 2575 . . . . . . . . 9  |-  F/_ d { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
203, 19nfcxfr 2568 . . . . . . . 8  |-  F/_ d H
2120nfuni 4013 . . . . . . 7  |-  F/_ d U. H
222, 21nfcxfr 2568 . . . . . 6  |-  F/_ d P
23 nfcv 2571 . . . . . . . 8  |-  F/_ d
x
24 nfcv 2571 . . . . . . . . 9  |-  F/_ d G
25 bnj1446.11 . . . . . . . . . 10  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2622, 4nfres 5140 . . . . . . . . . . 11  |-  F/_ d
( P  |`  pred (
x ,  A ,  R ) )
2723, 26nfop 3992 . . . . . . . . . 10  |-  F/_ d <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2825, 27nfcxfr 2568 . . . . . . . . 9  |-  F/_ d Z
2924, 28nffv 5727 . . . . . . . 8  |-  F/_ d
( G `  Z
)
3023, 29nfop 3992 . . . . . . 7  |-  F/_ d <. x ,  ( G `
 Z ) >.
3130nfsn 3858 . . . . . 6  |-  F/_ d { <. x ,  ( G `  Z )
>. }
3222, 31nfun 3495 . . . . 5  |-  F/_ d
( P  u.  { <. x ,  ( G `
 Z ) >. } )
331, 32nfcxfr 2568 . . . 4  |-  F/_ d Q
34 nfcv 2571 . . . 4  |-  F/_ d
z
3533, 34nffv 5727 . . 3  |-  F/_ d
( Q `  z
)
36 bnj1446.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
37 nfcv 2571 . . . . . . 7  |-  F/_ d  pred ( z ,  A ,  R )
3833, 37nfres 5140 . . . . . 6  |-  F/_ d
( Q  |`  pred (
z ,  A ,  R ) )
3934, 38nfop 3992 . . . . 5  |-  F/_ d <. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
4036, 39nfcxfr 2568 . . . 4  |-  F/_ d W
4124, 40nffv 5727 . . 3  |-  F/_ d
( G `  W
)
4235, 41nfeq 2578 . 2  |-  F/ d ( Q `  z
)  =  ( G `
 W )
4342nfri 1778 1  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   [.wsbc 3153    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   U.cuni 4007   class class class wbr 4204   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446    predc-bnj14 28989    FrSe w-bnj15 28993    trClc-bnj18 28995
This theorem is referenced by:  bnj1450  29356  bnj1463  29361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-res 4882  df-iota 5410  df-fv 5454
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