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Theorem bnj1452 29861
Description: Technical lemma for bnj60 29871. 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 29611 . . . . 5  |-  ( ch 
->  x  e.  A
)
54snssd 4117 . . . 4  |-  ( ch 
->  { x }  C_  A )
6 bnj1147 29803 . . . . 5  |-  trCl (
x ,  A ,  R )  C_  A
76a1i 11 . . . 4  |-  ( ch 
->  trCl ( x ,  A ,  R ) 
C_  A )
85, 7unssd 3610 . . 3  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  C_  A )
91, 8syl5eqss 3476 . 2  |-  ( ch 
->  E  C_  A )
10 elsni 3993 . . . . . . . 8  |-  ( z  e.  { x }  ->  z  =  x )
1110adantl 468 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  z  =  x )
12 bnj602 29726 . . . . . . 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 462 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
163, 15bnj835 29570 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
17 bnj906 29741 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1816, 4, 17syl2anc 667 . . . . . . 7  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1918ad2antrr 732 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
2013, 19eqsstrd 3466 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
21 ssun4 3600 . . . . . 6  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2221, 1syl6sseqr 3479 . . . . 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 732 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  R  FrSe  A )
25 simpr 463 . . . . . . . 8  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  trCl ( x ,  A ,  R ) )
266, 25bnj1213 29610 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  A )
27 bnj906 29741 . . . . . . 7  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
2824, 26, 27syl2anc 667 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( z ,  A ,  R
) )
294ad2antrr 732 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  x  e.  A )
30 bnj1125 29801 . . . . . . 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 1268 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  trCl ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3228, 31sstrd 3442 . . . . 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 29650 . . . . 5  |-  ( z  e.  E  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3534adantl 468 . . . 4  |-  ( ( ch  /\  z  e.  E )  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3623, 33, 35mpjaodan 795 . . 3  |-  ( ( ch  /\  z  e.  E )  ->  pred (
z ,  A ,  R )  C_  E
)
3736ralrimiva 2802 . 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 29739 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  e.  _V )
4116, 4, 40syl2anc 667 . . . . . . 7  |-  ( ch 
->  trCl ( x ,  A ,  R )  e.  _V )
4239, 41bnj1149 29604 . . . . . 6  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  e.  _V )
431, 42syl5eqel 2533 . . . . 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 29653 . . . . 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 29726 . . . . . . . 8  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
4847sseq1d 3459 . . . . . . 7  |-  ( x  =  z  ->  (  pred ( x ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  d
) )
4948cbvralv 3019 . . . . . 6  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  <->  A. z  e.  d  pred ( z ,  A ,  R )  C_  d
)
5049anbi2i 700 . . . . 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 3323 . . . 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 265 . . 3  |-  ( ch 
->  ( E  e.  B  <->  [. E  /  d ]. ( d  C_  A  /\  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d ) ) )
53 sseq1 3453 . . . . . 6  |-  ( d  =  E  ->  (
d  C_  A  <->  E  C_  A
) )
54 sseq2 3454 . . . . . . 7  |-  ( d  =  E  ->  (  pred ( z ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  E
) )
5554raleqbi1dv 2995 . . . . . 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 717 . . . . 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 3300 . . . 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 257 . 2  |-  ( ch 
->  ( E  e.  B  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
) ) )
609, 37, 59mpbir2and 933 1  |-  ( ch 
->  E  e.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   {cab 2437    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045   [.wsbc 3267    u. cun 3402    C_ wss 3404   (/)c0 3731   {csn 3968   <.cop 3974   U.cuni 4198   class class class wbr 4402   dom cdm 4834    |` cres 4836    Fn wfn 5577   ` cfv 5582    predc-bnj14 29493    FrSe w-bnj15 29497    trClc-bnj18 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-bnj17 29492  df-bnj14 29494  df-bnj13 29496  df-bnj15 29498  df-bnj18 29500  df-bnj19 29502
This theorem is referenced by:  bnj1312  29867
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