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Theorem bnj1442 32038
Description: Technical lemma for bnj60 32051. 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 31761 . . . . 5  |-  ( ch 
->  Fun  Q )
5 opex 4555 . . . . . . . 8  |-  <. x ,  ( G `  Z ) >.  e.  _V
65snid 3904 . . . . . . 7  |-  <. x ,  ( G `  Z ) >.  e.  { <. x ,  ( G `
 Z ) >. }
7 elun2 3523 . . . . . . 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 2515 . . . . 5  |-  <. x ,  ( G `  Z ) >.  e.  Q
11 funopfv 5730 . . . . 5  |-  ( Fun 
Q  ->  ( <. x ,  ( G `  Z ) >.  e.  Q  ->  ( Q `  x
)  =  ( G `
 Z ) ) )
124, 10, 11mpisyl 18 . . . 4  |-  ( ch 
->  ( Q `  x
)  =  ( G `
 Z ) )
132, 12bnj832 31748 . . 3  |-  ( th 
->  ( Q `  x
)  =  ( G `
 Z ) )
141, 13bnj832 31748 . 2  |-  ( et 
->  ( Q `  x
)  =  ( G `
 Z ) )
15 elsni 3901 . . . 4  |-  ( z  e.  { x }  ->  z  =  x )
161, 15bnj833 31749 . . 3  |-  ( et 
->  z  =  x
)
1716fveq2d 5694 . 2  |-  ( et 
->  ( Q `  z
)  =  ( Q `
 x ) )
18 bnj602 31906 . . . . . . . 8  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1918reseq2d 5109 . . . . . . 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 31762 . . . . . . . . . 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 31750 . . . . . . . . . . 11  |-  ( ch 
->  R  FrSe  A )
27 bnj1442.5 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  -.  E. f ta }
2827, 23bnj1212 31791 . . . . . . . . . . 11  |-  ( ch 
->  x  e.  A
)
29 bnj906 31921 . . . . . . . . . . 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 5509 . . . . . . . . . . 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 3392 . . . . . . . . 9  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  P )
354, 22, 34bnj1503 31840 . . . . . . . 8  |-  ( ch 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
362, 35bnj832 31748 . . . . . . 7  |-  ( th 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
371, 36bnj832 31748 . . . . . 6  |-  ( et 
->  ( Q  |`  pred (
x ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3820, 37eqtrd 2474 . . . . 5  |-  ( et 
->  ( Q  |`  pred (
z ,  A ,  R ) )  =  ( P  |`  pred (
x ,  A ,  R ) ) )
3916, 38opeq12d 4066 . . . 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 2499 . . 3  |-  ( et 
->  W  =  Z
)
4342fveq2d 5694 . 2  |-  ( et 
->  ( G `  W
)  =  ( G `
 Z ) )
4414, 17, 433eqtr4d 2484 1  |-  ( et 
->  ( Q `  z
)  =  ( G `
 W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2428    =/= wne 2605   A.wral 2714   E.wrex 2715   {crab 2718   [.wsbc 3185    u. cun 3325    C_ wss 3327   (/)c0 3636   {csn 3876   <.cop 3882   U.cuni 4090   class class class wbr 4291   dom cdm 4839    |` cres 4841   Fun wfun 5411    Fn wfn 5412   ` cfv 5417    predc-bnj14 31674    FrSe w-bnj15 31678    trClc-bnj18 31680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-reg 7806  ax-inf2 7846
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-om 6476  df-1o 6919  df-bnj17 31673  df-bnj14 31675  df-bnj13 31677  df-bnj15 31679  df-bnj18 31681
This theorem is referenced by:  bnj1423  32040
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