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Theorem bnj1446 28753
Description: Technical lemma for bnj60 28770. 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 2524 . . . . . . . . . . 11  |-  F/_ d  pred ( x ,  A ,  R )
5 bnj1446.8 . . . . . . . . . . . 12  |-  ( ta'  <->  [. y  /  x ]. ta )
6 nfcv 2524 . . . . . . . . . . . . 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 2706 . . . . . . . . . . . . . . . . . 18  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
109nfab 2528 . . . . . . . . . . . . . . . . 17  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
118, 10nfcxfr 2521 . . . . . . . . . . . . . . . 16  |-  F/_ d C
1211nfcri 2518 . . . . . . . . . . . . . . 15  |-  F/ d  f  e.  C
13 nfv 1626 . . . . . . . . . . . . . . 15  |-  F/ d dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
1412, 13nfan 1836 . . . . . . . . . . . . . 14  |-  F/ d ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
157, 14nfxfr 1576 . . . . . . . . . . . . 13  |-  F/ d ta
166, 15nfsbc 3126 . . . . . . . . . . . 12  |-  F/ d
[. y  /  x ]. ta
175, 16nfxfr 1576 . . . . . . . . . . 11  |-  F/ d ta'
184, 17nfrex 2705 . . . . . . . . . 10  |-  F/ d E. y  e.  pred  ( x ,  A ,  R ) ta'
1918nfab 2528 . . . . . . . . 9  |-  F/_ d { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
203, 19nfcxfr 2521 . . . . . . . 8  |-  F/_ d H
2120nfuni 3964 . . . . . . 7  |-  F/_ d U. H
222, 21nfcxfr 2521 . . . . . 6  |-  F/_ d P
23 nfcv 2524 . . . . . . . 8  |-  F/_ d
x
24 nfcv 2524 . . . . . . . . 9  |-  F/_ d G
25 bnj1446.11 . . . . . . . . . 10  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2622, 4nfres 5089 . . . . . . . . . . 11  |-  F/_ d
( P  |`  pred (
x ,  A ,  R ) )
2723, 26nfop 3943 . . . . . . . . . 10  |-  F/_ d <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
2825, 27nfcxfr 2521 . . . . . . . . 9  |-  F/_ d Z
2924, 28nffv 5676 . . . . . . . 8  |-  F/_ d
( G `  Z
)
3023, 29nfop 3943 . . . . . . 7  |-  F/_ d <. x ,  ( G `
 Z ) >.
3130nfsn 3810 . . . . . 6  |-  F/_ d { <. x ,  ( G `  Z )
>. }
3222, 31nfun 3447 . . . . 5  |-  F/_ d
( P  u.  { <. x ,  ( G `
 Z ) >. } )
331, 32nfcxfr 2521 . . . 4  |-  F/_ d Q
34 nfcv 2524 . . . 4  |-  F/_ d
z
3533, 34nffv 5676 . . 3  |-  F/_ d
( Q `  z
)
36 bnj1446.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
37 nfcv 2524 . . . . . . 7  |-  F/_ d  pred ( z ,  A ,  R )
3833, 37nfres 5089 . . . . . 6  |-  F/_ d
( Q  |`  pred (
z ,  A ,  R ) )
3934, 38nfop 3943 . . . . 5  |-  F/_ d <. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
4036, 39nfcxfr 2521 . . . 4  |-  F/_ d W
4124, 40nffv 5676 . . 3  |-  F/_ d
( G `  W
)
4235, 41nfeq 2531 . 2  |-  F/ d ( Q `  z
)  =  ( G `
 W )
4342nfri 1770 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 1546   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374    =/= wne 2551   A.wral 2650   E.wrex 2651   {crab 2654   [.wsbc 3105    u. cun 3262    C_ wss 3264   (/)c0 3572   {csn 3758   <.cop 3761   U.cuni 3958   class class class wbr 4154   dom cdm 4819    |` cres 4821    Fn wfn 5390   ` cfv 5395    predc-bnj14 28391    FrSe w-bnj15 28395    trClc-bnj18 28397
This theorem is referenced by:  bnj1450  28758  bnj1463  28763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-res 4831  df-iota 5359  df-fv 5403
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