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Theorem bnj1452 29398
Description: Technical lemma for bnj60 29408. 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 29148 . . . . 5  |-  ( ch 
->  x  e.  A
)
54snssd 3776 . . . 4  |-  ( ch 
->  { x }  C_  A )
6 bnj1147 29340 . . . . 5  |-  trCl (
x ,  A ,  R )  C_  A
76a1i 10 . . . 4  |-  ( ch 
->  trCl ( x ,  A ,  R ) 
C_  A )
85, 7unssd 3364 . . 3  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  C_  A )
91, 8syl5eqss 3235 . 2  |-  ( ch 
->  E  C_  A )
10 elsni 3677 . . . . . . . 8  |-  ( z  e.  { x }  ->  z  =  x )
1110adantl 452 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  z  =  x )
12 bnj602 29263 . . . . . . 7  |-  ( z  =  x  ->  pred (
z ,  A ,  R )  =  pred ( x ,  A ,  R ) )
1311, 12syl 15 . . . . . 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 446 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
163, 15bnj835 29105 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
17 bnj906 29278 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1816, 4, 17syl2anc 642 . . . . . . 7  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
1918ad2antrr 706 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
2013, 19eqsstrd 3225 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
21 ssun4 3354 . . . . . 6  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2221, 1syl6sseqr 3238 . . . . 5  |-  (  pred ( z ,  A ,  R )  C_  trCl (
x ,  A ,  R )  ->  pred (
z ,  A ,  R )  C_  E
)
2320, 22syl 15 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  pred ( z ,  A ,  R ) 
C_  E )
2416ad2antrr 706 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  R  FrSe  A )
25 simpr 447 . . . . . . . 8  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  trCl ( x ,  A ,  R ) )
266, 25bnj1213 29147 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  z  e.  A )
27 bnj906 29278 . . . . . . 7  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
2824, 26, 27syl2anc 642 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( z ,  A ,  R
) )
294ad2antrr 706 . . . . . . 7  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  x  e.  A )
30 bnj1125 29338 . . . . . . 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 1182 . . . . . 6  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  trCl ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3228, 31sstrd 3202 . . . . 5  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3332, 22syl 15 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  pred ( z ,  A ,  R
)  C_  E )
341bnj1424 29187 . . . . 5  |-  ( z  e.  E  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3534adantl 452 . . . 4  |-  ( ( ch  /\  z  e.  E )  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
3623, 33, 35mpjaodan 761 . . 3  |-  ( ( ch  /\  z  e.  E )  ->  pred (
z ,  A ,  R )  C_  E
)
3736ralrimiva 2639 . 2  |-  ( ch 
->  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
)
38 snex 4232 . . . . . . 7  |-  { x }  e.  _V
3938a1i 10 . . . . . 6  |-  ( ch 
->  { x }  e.  _V )
40 bnj893 29276 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  e.  _V )
4116, 4, 40syl2anc 642 . . . . . 6  |-  ( ch 
->  trCl ( x ,  A ,  R )  e.  _V )
4239, 41bnj1149 29140 . . . . 5  |-  ( ch 
->  ( { x }  u.  trCl ( x ,  A ,  R ) )  e.  _V )
431, 42syl5eqel 2380 . . . 4  |-  ( ch 
->  E  e.  _V )
44 bnj1452.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
4544bnj1454 29190 . . . 4  |-  ( E  e.  _V  ->  ( E  e.  B  <->  [. E  / 
d ]. ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R ) 
C_  d ) ) )
4643, 45syl 15 . . 3  |-  ( ch 
->  ( E  e.  B  <->  [. E  /  d ]. ( d  C_  A  /\  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) ) )
47 bnj602 29263 . . . . . . . 8  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
4847sseq1d 3218 . . . . . . 7  |-  ( x  =  z  ->  (  pred ( x ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  d
) )
4948cbvralv 2777 . . . . . 6  |-  ( A. x  e.  d  pred ( x ,  A ,  R )  C_  d  <->  A. z  e.  d  pred ( z ,  A ,  R )  C_  d
)
5049anbi2i 675 . . . . 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 ) )
5150sbcbiiOLD 3060 . . . 4  |-  ( E  e.  _V  ->  ( [. 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 ) ) )
5243, 51syl 15 . . 3  |-  ( ch 
->  ( [. 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 ) ) )
53 sseq1 3212 . . . . . 6  |-  ( d  =  E  ->  (
d  C_  A  <->  E  C_  A
) )
54 sseq2 3213 . . . . . . 7  |-  ( d  =  E  ->  (  pred ( z ,  A ,  R )  C_  d  <->  pred ( z ,  A ,  R )  C_  E
) )
5554raleqbi1dv 2757 . . . . . 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 691 . . . . 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 3036 . . . 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 15 . . 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 )
) )
5946, 52, 583bitrd 270 . 2  |-  ( ch 
->  ( E  e.  B  <->  ( E  C_  A  /\  A. z  e.  E  pred ( z ,  A ,  R )  C_  E
) ) )
609, 37, 59mpbir2and 888 1  |-  ( ch 
->  E  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   [.wsbc 3004    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U.cuni 3843   class class class wbr 4039   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1312  29404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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