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Theorem bnj1442 33540
Description: Technical lemma for bnj60 33553. 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
bnj1442.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1442.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1442.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1442.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1442.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1442.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1442.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1442.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1442.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1442.10  |-  P  = 
U. H
bnj1442.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1442.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1442.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1442.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1442.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1442.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1442.17  |-  ( th  <->  ( ch  /\  z  e.  E ) )
bnj1442.18  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
Assertion
Ref Expression
bnj1442  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Distinct variable group:    x, A
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, z, f, d)    th( x, y, z, f, d)    ta( x, y, z, f, d)    et( x, y, z, f, d)    A( y, 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, y, z, f, d)    E( x, y, z, f, d)    G( x, y, 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 bnj1442
StepHypRef Expression
1 bnj1442.18 . . 3  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
2 bnj1442.17 . . . 4  |-  ( th  <->  ( ch  /\  z  e.  E ) )
3 bnj1442.16 . . . . . 6  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
43bnj930 33263 . . . . 5  |-  ( ch 
->  Fun  Q )
5 opex 4717 . . . . . . . 8  |-  <. x ,  ( G `  Z ) >.  e.  _V
65snid 4061 . . . . . . 7  |-  <. x ,  ( G `  Z ) >.  e.  { <. x ,  ( G `
 Z ) >. }
7 elun2 3677 . . . . . . 7  |-  ( <.
x ,  ( G `
 Z ) >.  e.  { <. x ,  ( G `  Z )
>. }  ->  <. x ,  ( G `  Z
) >.  e.  ( P  u.  { <. x ,  ( G `  Z ) >. } ) )
86, 7ax-mp 5 . . . . . 6  |-  <. x ,  ( G `  Z ) >.  e.  ( P  u.  { <. x ,  ( G `  Z ) >. } )
9 bnj1442.12 . . . . . 6  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
108, 9eleqtrri 2554 . . . . 5  |-  <. x ,  ( G `  Z ) >.  e.  Q
11 funopfv 5913 . . . . 5  |-  ( Fun 
Q  ->  ( <. x ,  ( G `  Z ) >.  e.  Q  ->  ( Q `  x
)  =  ( G `
 Z ) ) )
124, 10, 11mpisyl 18 . . . 4  |-  ( ch 
->  ( Q `  x
)  =  ( G `
 Z ) )
132, 12bnj832 33250 . . 3  |-  ( th 
->  ( Q `  x
)  =  ( G `
 Z ) )
141, 13bnj832 33250 . 2  |-  ( et 
->  ( Q `  x
)  =  ( G `
 Z ) )
15 elsni 4058 . . . 4  |-  ( z  e.  { x }  ->  z  =  x )
161, 15bnj833 33251 . . 3  |-  ( et 
->  z  =  x
)
1716fveq2d 5876 . 2  |-  ( et 
->  ( Q `  z
)  =  ( Q `
 x ) )
18 bnj602 33408 . . . . . . . 8  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1918reseq2d 5279 . . . . . . 7  |-  ( z  =  x  ->  ( Q  |`  pred ( z ,  A ,  R ) )  =  ( Q  |`  pred ( x ,  A ,  R ) ) )
2016, 19syl 16 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( Q  |`  pred (
x ,  A ,  R ) ) )
219bnj931 33264 . . . . . . . . . 10  |-  P  C_  Q
2221a1i 11 . . . . . . . . 9  |-  ( ch 
->  P  C_  Q )
23 bnj1442.7 . . . . . . . . . . . 12  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
24 bnj1442.6 . . . . . . . . . . . . 13  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
2524simplbi 460 . . . . . . . . . . . 12  |-  ( ps 
->  R  FrSe  A )
2623, 25bnj835 33252 . . . . . . . . . . 11  |-  ( ch 
->  R  FrSe  A )
27 bnj1442.5 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  -.  E. f ta }
2827, 23bnj1212 33293 . . . . . . . . . . 11  |-  ( ch 
->  x  e.  A
)
29 bnj906 33423 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3026, 28, 29syl2anc 661 . . . . . . . . . 10  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
31 bnj1442.15 . . . . . . . . . . 11  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
32 fndm 5686 . . . . . . . . . . 11  |-  ( P  Fn  trCl ( x ,  A ,  R )  ->  dom  P  =  trCl ( x ,  A ,  R ) )
3331, 32syl 16 . . . . . . . . . 10  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
3430, 33sseqtr4d 3546 . . . . . . . . 9  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  P )
354, 22, 34bnj1503 33342 . . . . . . . 8  |-  ( ch 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
362, 35bnj832 33250 . . . . . . 7  |-  ( th 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
371, 36bnj832 33250 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3820, 37eqtrd 2508 . . . . 5  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3916, 38opeq12d 4227 . . . 4  |-  ( et 
->  <. z ,  ( Q  |`  pred ( z ,  A ,  R
) ) >.  =  <. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
)
40 bnj1442.13 . . . 4  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
41 bnj1442.11 . . . 4  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
4239, 40, 413eqtr4g 2533 . . 3  |-  ( et 
->  W  =  Z
)
4342fveq2d 5876 . 2  |-  ( et 
->  ( G `  W
)  =  ( G `
 Z ) )
4414, 17, 433eqtr4d 2518 1  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   [.wsbc 3336    u. cun 3479    C_ wss 3481   (/)c0 3790   {csn 4033   <.cop 4039   U.cuni 4251   class class class wbr 4453   dom cdm 5005    |` cres 5007   Fun wfun 5588    Fn wfn 5589   ` cfv 5594    predc-bnj14 33176    FrSe w-bnj15 33180    trClc-bnj18 33182
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-bnj17 33175  df-bnj14 33177  df-bnj13 33179  df-bnj15 33181  df-bnj18 33183
This theorem is referenced by:  bnj1423  33542
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