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Theorem bnj1452 34209
Description: Technical lemma for bnj60 34219. 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 33959 . . . . 5  |-  ( ch 
->  x  e.  A
)
54snssd 4177 . . . 4  |-  ( ch 
->  { x }  C_  A )
6 bnj1147 34151 . . . . 5  |-  trCl (
x ,  A ,  R )  C_  A
76a1i 11 . . . 4  |-  ( ch 
->  trCl ( x ,  A ,  R ) 
C_  A )
85, 7unssd 3676 . . 3  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  C_  A )
91, 8syl5eqss 3543 . 2  |-  ( ch 
->  E  C_  A )
10 elsni 4057 . . . . . . . 8  |-  ( z  e.  { x }  ->  z  =  x )
1110adantl 466 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  z  =  x )
12 bnj602 34074 . . . . . . 7  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1311, 12syl 16 . . . . . 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 460 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
163, 15bnj835 33918 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
17 bnj906 34089 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1816, 4, 17syl2anc 661 . . . . . . 7  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1918ad2antrr 725 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
2013, 19eqsstrd 3533 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
21 ssun4 3666 . . . . . 6  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2221, 1syl6sseqr 3546 . . . . 5  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  E
)
2320, 22syl 16 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  E )
2416ad2antrr 725 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  R  FrSe  A )
25 simpr 461 . . . . . . . 8  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  trCl ( x ,  A ,  R ) )
266, 25bnj1213 33958 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  A )
27 bnj906 34089 . . . . . . 7  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
2824, 26, 27syl2anc 661 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( z ,  A ,  R
) )
294ad2antrr 725 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  x  e.  A )
30 bnj1125 34149 . . . . . . 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 1228 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  trCl ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3228, 31sstrd 3509 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3332, 22syl 16 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  E )
341bnj1424 33998 . . . . 5  |-  ( z  e.  E  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3534adantl 466 . . . 4  |-  ( ( ch  /\  z  e.  E )  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3623, 33, 35mpjaodan 786 . . 3  |-  ( ( ch  /\  z  e.  E )  ->  pred (
z ,  A ,  R )  C_  E
)
3736ralrimiva 2871 . 2  |-  ( ch 
->  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
)
38 snex 4697 . . . . . . . 8  |-  { x }  e.  _V
3938a1i 11 . . . . . . 7  |-  ( ch 
->  { x }  e.  _V )
40 bnj893 34087 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  e.  _V )
4116, 4, 40syl2anc 661 . . . . . . 7  |-  ( ch 
->  trCl ( x ,  A ,  R )  e.  _V )
4239, 41bnj1149 33952 . . . . . 6  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  e.  _V )
431, 42syl5eqel 2549 . . . . 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 34001 . . . . 5  |-  ( E  e.  _V  ->  ( E  e.  B  <->  [. E  / 
d ]. ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R ) 
C_  d ) ) )
4643, 45syl 16 . . . 4  |-  ( ch 
->  ( E  e.  B  <->  [. E  /  d ]. ( d  C_  A  /\  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) ) )
47 bnj602 34074 . . . . . . . 8  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
4847sseq1d 3526 . . . . . . 7  |-  ( x  =  z  ->  (  pred ( x ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  d
) )
4948cbvralv 3084 . . . . . 6  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  <->  A. z  e.  d  pred ( z ,  A ,  R )  C_  d
)
5049anbi2i 694 . . . . 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 3387 . . . 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 261 . . 3  |-  ( ch 
->  ( E  e.  B  <->  [. E  /  d ]. ( d  C_  A  /\  A. z  e.  d 
pred ( z ,  A ,  R ) 
C_  d ) ) )
53 sseq1 3520 . . . . . 6  |-  ( d  =  E  ->  (
d  C_  A  <->  E  C_  A
) )
54 sseq2 3521 . . . . . . 7  |-  ( d  =  E  ->  (  pred ( z ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  E
) )
5554raleqbi1dv 3062 . . . . . 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 710 . . . . 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 3360 . . . 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 16 . . 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 253 . 2  |-  ( ch 
->  ( E  e.  B  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
) ) )
609, 37, 59mpbir2and 922 1  |-  ( ch 
->  E  e.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109   [.wsbc 3327    u. cun 3469    C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038   U.cuni 4251   class class class wbr 4456   dom cdm 5008    |` cres 5010    Fn wfn 5589   ` cfv 5594    predc-bnj14 33841    FrSe w-bnj15 33845    trClc-bnj18 33847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6700  df-1o 7148  df-bnj17 33840  df-bnj14 33842  df-bnj13 33844  df-bnj15 33846  df-bnj18 33848  df-bnj19 33850
This theorem is referenced by:  bnj1312  34215
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