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Theorem bnj1442 29124
Description: Technical lemma for bnj60 29137. 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 28846 . . . . 5  |-  ( ch 
->  Fun  Q )
5 opex 4387 . . . . . . . 8  |-  <. x ,  ( G `  Z ) >.  e.  _V
65snid 3801 . . . . . . 7  |-  <. x ,  ( G `  Z ) >.  e.  { <. x ,  ( G `
 Z ) >. }
7 elun2 3475 . . . . . . 7  |-  ( <.
x ,  ( G `
 Z ) >.  e.  { <. x ,  ( G `  Z )
>. }  ->  <. x ,  ( G `  Z
) >.  e.  ( P  u.  { <. x ,  ( G `  Z ) >. } ) )
86, 7ax-mp 8 . . . . . 6  |-  <. x ,  ( G `  Z ) >.  e.  ( P  u.  { <. x ,  ( G `  Z ) >. } )
9 bnj1442.12 . . . . . 6  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
108, 9eleqtrri 2477 . . . . 5  |-  <. x ,  ( G `  Z ) >.  e.  Q
11 funopfv 5725 . . . . 5  |-  ( Fun 
Q  ->  ( <. x ,  ( G `  Z ) >.  e.  Q  ->  ( Q `  x
)  =  ( G `
 Z ) ) )
124, 10, 11ee10 1382 . . . 4  |-  ( ch 
->  ( Q `  x
)  =  ( G `
 Z ) )
132, 12bnj832 28832 . . 3  |-  ( th 
->  ( Q `  x
)  =  ( G `
 Z ) )
141, 13bnj832 28832 . 2  |-  ( et 
->  ( Q `  x
)  =  ( G `
 Z ) )
15 elsni 3798 . . . 4  |-  ( z  e.  { x }  ->  z  =  x )
161, 15bnj833 28833 . . 3  |-  ( et 
->  z  =  x
)
1716fveq2d 5691 . 2  |-  ( et 
->  ( Q `  z
)  =  ( Q `
 x ) )
18 bnj602 28992 . . . . . . . 8  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1918reseq2d 5105 . . . . . . 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 28847 . . . . . . . . . 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 447 . . . . . . . . . . . 12  |-  ( ps 
->  R  FrSe  A )
2623, 25bnj835 28834 . . . . . . . . . . 11  |-  ( ch 
->  R  FrSe  A )
27 bnj1442.5 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  -.  E. f ta }
2827, 23bnj1212 28877 . . . . . . . . . . 11  |-  ( ch 
->  x  e.  A
)
29 bnj906 29007 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3026, 28, 29syl2anc 643 . . . . . . . . . 10  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
31 bnj1442.15 . . . . . . . . . . 11  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
32 fndm 5503 . . . . . . . . . . 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 3345 . . . . . . . . 9  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  P )
354, 22, 34bnj1503 28926 . . . . . . . 8  |-  ( ch 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
362, 35bnj832 28832 . . . . . . 7  |-  ( th 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
371, 36bnj832 28832 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3820, 37eqtrd 2436 . . . . 5  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3916, 38opeq12d 3952 . . . 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 2461 . . 3  |-  ( et 
->  W  =  Z
)
4342fveq2d 5691 . 2  |-  ( et 
->  ( G `  W
)  =  ( G `
 Z ) )
4414, 17, 433eqtr4d 2446 1  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   [.wsbc 3121    u. cun 3278    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   U.cuni 3975   class class class wbr 4172   dom cdm 4837    |` cres 4839   Fun wfun 5407    Fn wfn 5408   ` cfv 5413    predc-bnj14 28758    FrSe w-bnj15 28762    trClc-bnj18 28764
This theorem is referenced by:  bnj1423  29126
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-bnj17 28757  df-bnj14 28759  df-bnj13 28761  df-bnj15 28763  df-bnj18 28765
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