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Theorem bnj1452 29933
Description: Technical lemma for bnj60 29943. 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
bnj1452.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1452.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1452.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1452.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1452.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1452.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1452.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1452.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1452.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1452.10  |-  P  = 
U. H
bnj1452.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1452.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1452.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1452.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj1452  |-  ( ch 
->  E  e.  B
)
Distinct variable groups:    A, d, x, z    E, d, z    R, d, x, z    ch, z
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, f, d)    ta( x, y, z, f, d)    A( y, f)    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( y, f)    E( x, y, f)    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 bnj1452
StepHypRef Expression
1 bnj1452.14 . . 3  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
2 bnj1452.5 . . . . . 6  |-  D  =  { x  e.  A  |  -.  E. f ta }
3 bnj1452.7 . . . . . 6  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
42, 3bnj1212 29683 . . . . 5  |-  ( ch 
->  x  e.  A
)
54snssd 4108 . . . 4  |-  ( ch 
->  { x }  C_  A )
6 bnj1147 29875 . . . . 5  |-  trCl (
x ,  A ,  R )  C_  A
76a1i 11 . . . 4  |-  ( ch 
->  trCl ( x ,  A ,  R ) 
C_  A )
85, 7unssd 3601 . . 3  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  C_  A )
91, 8syl5eqss 3462 . 2  |-  ( ch 
->  E  C_  A )
10 elsni 3985 . . . . . . . 8  |-  ( z  e.  { x }  ->  z  =  x )
1110adantl 473 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  z  =  x )
12 bnj602 29798 . . . . . . 7  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1311, 12syl 17 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
14 bnj1452.6 . . . . . . . . . 10  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
1514simplbi 467 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
163, 15bnj835 29642 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
17 bnj906 29813 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1816, 4, 17syl2anc 673 . . . . . . 7  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1918ad2antrr 740 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
2013, 19eqsstrd 3452 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
21 ssun4 3591 . . . . . 6  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2221, 1syl6sseqr 3465 . . . . 5  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  E
)
2320, 22syl 17 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  E )
2416ad2antrr 740 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  R  FrSe  A )
25 simpr 468 . . . . . . . 8  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  trCl ( x ,  A ,  R ) )
266, 25bnj1213 29682 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  A )
27 bnj906 29813 . . . . . . 7  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
2824, 26, 27syl2anc 673 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( z ,  A ,  R
) )
294ad2antrr 740 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  x  e.  A )
30 bnj1125 29873 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  z  e.  trCl ( x ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl (
x ,  A ,  R ) )
3124, 29, 25, 30syl3anc 1292 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  trCl ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3228, 31sstrd 3428 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3332, 22syl 17 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  E )
341bnj1424 29722 . . . . 5  |-  ( z  e.  E  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3534adantl 473 . . . 4  |-  ( ( ch  /\  z  e.  E )  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3623, 33, 35mpjaodan 803 . . 3  |-  ( ( ch  /\  z  e.  E )  ->  pred (
z ,  A ,  R )  C_  E
)
3736ralrimiva 2809 . 2  |-  ( ch 
->  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
)
38 snex 4641 . . . . . . . 8  |-  { x }  e.  _V
3938a1i 11 . . . . . . 7  |-  ( ch 
->  { x }  e.  _V )
40 bnj893 29811 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  e.  _V )
4116, 4, 40syl2anc 673 . . . . . . 7  |-  ( ch 
->  trCl ( x ,  A ,  R )  e.  _V )
4239, 41bnj1149 29676 . . . . . 6  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  e.  _V )
431, 42syl5eqel 2553 . . . . 5  |-  ( ch 
->  E  e.  _V )
44 bnj1452.1 . . . . . 6  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
4544bnj1454 29725 . . . . 5  |-  ( E  e.  _V  ->  ( E  e.  B  <->  [. E  / 
d ]. ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R ) 
C_  d ) ) )
4643, 45syl 17 . . . 4  |-  ( ch 
->  ( E  e.  B  <->  [. E  /  d ]. ( d  C_  A  /\  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) ) )
47 bnj602 29798 . . . . . . . 8  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
4847sseq1d 3445 . . . . . . 7  |-  ( x  =  z  ->  (  pred ( x ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  d
) )
4948cbvralv 3005 . . . . . 6  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  <->  A. z  e.  d  pred ( z ,  A ,  R )  C_  d
)
5049anbi2i 708 . . . . 5  |-  ( ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
)  <->  ( d  C_  A  /\  A. z  e.  d  pred ( z ,  A ,  R ) 
C_  d ) )
5150sbcbii 3311 . . . 4  |-  ( [. E  /  d ]. (
d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
)  <->  [. E  /  d ]. ( d  C_  A  /\  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d ) )
5246, 51syl6bb 269 . . 3  |-  ( ch 
->  ( E  e.  B  <->  [. E  /  d ]. ( d  C_  A  /\  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d ) ) )
53 sseq1 3439 . . . . . 6  |-  ( d  =  E  ->  (
d  C_  A  <->  E  C_  A
) )
54 sseq2 3440 . . . . . . 7  |-  ( d  =  E  ->  (  pred ( z ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  E
) )
5554raleqbi1dv 2981 . . . . . 6  |-  ( d  =  E  ->  ( A. z  e.  d  pred ( z ,  A ,  R )  C_  d  <->  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
) )
5653, 55anbi12d 725 . . . . 5  |-  ( d  =  E  ->  (
( d  C_  A  /\  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d )  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R
)  C_  E )
) )
5756sbcieg 3288 . . . 4  |-  ( E  e.  _V  ->  ( [. E  /  d ]. ( d  C_  A  /\  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d )  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R
)  C_  E )
) )
5843, 57syl 17 . . 3  |-  ( ch 
->  ( [. E  / 
d ]. ( d  C_  A  /\  A. z  e.  d  pred ( z ,  A ,  R ) 
C_  d )  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R
)  C_  E )
) )
5952, 58bitrd 261 . 2  |-  ( ch 
->  ( E  e.  B  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
) ) )
609, 37, 59mpbir2and 936 1  |-  ( ch 
->  E  e.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   [.wsbc 3255    u. cun 3388    C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395   dom cdm 4839    |` cres 4841    Fn wfn 5584   ` cfv 5589    predc-bnj14 29565    FrSe w-bnj15 29569    trClc-bnj18 29571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-bnj17 29564  df-bnj14 29566  df-bnj13 29568  df-bnj15 29570  df-bnj18 29572  df-bnj19 29574
This theorem is referenced by:  bnj1312  29939
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