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Theorem bnj1447 28840
Description: Technical lemma for bnj60 28856. 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 2684 . . . . . . . . . 10  |-  F/ y E. y  e.  pred  ( x ,  A ,  R ) ta'
54nfab 2506 . . . . . . . . 9  |-  F/_ y { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
63, 5nfcxfr 2499 . . . . . . . 8  |-  F/_ y H
76nfuni 3935 . . . . . . 7  |-  F/_ y U. H
82, 7nfcxfr 2499 . . . . . 6  |-  F/_ y P
9 nfcv 2502 . . . . . . . 8  |-  F/_ y
x
10 nfcv 2502 . . . . . . . . 9  |-  F/_ y G
11 bnj1447.11 . . . . . . . . . 10  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
12 nfcv 2502 . . . . . . . . . . . 12  |-  F/_ y  pred ( x ,  A ,  R )
138, 12nfres 5060 . . . . . . . . . . 11  |-  F/_ y
( P  |`  pred (
x ,  A ,  R ) )
149, 13nfop 3914 . . . . . . . . . 10  |-  F/_ y <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
1511, 14nfcxfr 2499 . . . . . . . . 9  |-  F/_ y Z
1610, 15nffv 5639 . . . . . . . 8  |-  F/_ y
( G `  Z
)
179, 16nfop 3914 . . . . . . 7  |-  F/_ y <. x ,  ( G `
 Z ) >.
1817nfsn 3781 . . . . . 6  |-  F/_ y { <. x ,  ( G `  Z )
>. }
198, 18nfun 3419 . . . . 5  |-  F/_ y
( P  u.  { <. x ,  ( G `
 Z ) >. } )
201, 19nfcxfr 2499 . . . 4  |-  F/_ y Q
21 nfcv 2502 . . . 4  |-  F/_ y
z
2220, 21nffv 5639 . . 3  |-  F/_ y
( Q `  z
)
23 bnj1447.13 . . . . 5  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
24 nfcv 2502 . . . . . . 7  |-  F/_ y  pred ( z ,  A ,  R )
2520, 24nfres 5060 . . . . . 6  |-  F/_ y
( Q  |`  pred (
z ,  A ,  R ) )
2621, 25nfop 3914 . . . . 5  |-  F/_ y <. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
2723, 26nfcxfr 2499 . . . 4  |-  F/_ y W
2810, 27nffv 5639 . . 3  |-  F/_ y
( G `  W
)
2922, 28nfeq 2509 . 2  |-  F/ y ( Q `  z
)  =  ( G `
 W )
3029nfri 1768 1  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. y
( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935   A.wal 1545   E.wex 1546    = wceq 1647    e. wcel 1715   {cab 2352    =/= wne 2529   A.wral 2628   E.wrex 2629   {crab 2632   [.wsbc 3077    u. cun 3236    C_ wss 3238   (/)c0 3543   {csn 3729   <.cop 3732   U.cuni 3929   class class class wbr 4125   dom cdm 4792    |` cres 4794    Fn wfn 5353   ` cfv 5358    predc-bnj14 28477    FrSe w-bnj15 28481    trClc-bnj18 28483
This theorem is referenced by:  bnj1450  28844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-res 4804  df-iota 5322  df-fv 5366
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