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Theorem bnj1489 29086
Description: Technical lemma for bnj60 29092. 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
bnj1489.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1489.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1489.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1489.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1489.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1489.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1489.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1489.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1489.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1489.10  |-  P  = 
U. H
bnj1489.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1489.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
Assertion
Ref Expression
bnj1489  |-  ( ch 
->  Q  e.  _V )
Distinct variable groups:    A, d,
f, x    y, A, f, x    B, f    y, D    G, d, f    R, d, f, x    y, R    ps, y    ta, y
Allowed substitution hints:    ps( x, f, d)    ch( x, y, f, d)    ta( x, f, d)    B( x, y, d)    C( x, y, f, d)    D( x, f, d)    P( x, y, f, d)    Q( x, y, f, d)    G( x, y)    H( x, y, f, d)    Y( x, y, f, d)    Z( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1489
StepHypRef Expression
1 bnj1489.12 . 2  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
2 bnj1489.10 . . . 4  |-  P  = 
U. H
3 bnj1489.7 . . . . . . . 8  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
4 bnj1489.6 . . . . . . . . 9  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
5 bnj1364 29058 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  R  Se  A )
6 df-bnj13 28716 . . . . . . . . . 10  |-  ( R  Se  A  <->  A. x  e.  A  pred ( x ,  A ,  R
)  e.  _V )
75, 6sylib 188 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  A. x  e.  A  pred ( x ,  A ,  R
)  e.  _V )
84, 7bnj832 28787 . . . . . . . 8  |-  ( ps 
->  A. x  e.  A  pred ( x ,  A ,  R )  e.  _V )
93, 8bnj835 28789 . . . . . . 7  |-  ( ch 
->  A. x  e.  A  pred ( x ,  A ,  R )  e.  _V )
10 bnj1489.5 . . . . . . . 8  |-  D  =  { x  e.  A  |  -.  E. f ta }
1110, 3bnj1212 28832 . . . . . . 7  |-  ( ch 
->  x  e.  A
)
129, 11bnj1294 28850 . . . . . 6  |-  ( ch 
->  pred ( x ,  A ,  R )  e.  _V )
13 nfv 1605 . . . . . . . . 9  |-  F/ y ps
14 nfv 1605 . . . . . . . . 9  |-  F/ y  x  e.  D
15 nfra1 2593 . . . . . . . . 9  |-  F/ y A. y  e.  D  -.  y R x
1613, 14, 15nf3an 1774 . . . . . . . 8  |-  F/ y ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
173, 16nfxfr 1557 . . . . . . 7  |-  F/ y ch
184simplbi 446 . . . . . . . . . . 11  |-  ( ps 
->  R  FrSe  A )
193, 18bnj835 28789 . . . . . . . . . 10  |-  ( ch 
->  R  FrSe  A )
2019adantr 451 . . . . . . . . 9  |-  ( ( ch  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  R  FrSe  A )
21 bnj1489.1 . . . . . . . . . . 11  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
22 bnj1489.2 . . . . . . . . . . 11  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
23 bnj1489.3 . . . . . . . . . . 11  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
24 bnj1489.4 . . . . . . . . . . 11  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
25 bnj1489.8 . . . . . . . . . . 11  |-  ( ta'  <->  [. y  /  x ]. ta )
2621, 22, 23, 24, 10, 4, 3, 25bnj1388 29063 . . . . . . . . . 10  |-  ( ch 
->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
2726r19.21bi 2641 . . . . . . . . 9  |-  ( ( ch  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  E. f ta' )
28 nfv 1605 . . . . . . . . . . . 12  |-  F/ x  R  FrSe  A
29 nfsbc1v 3010 . . . . . . . . . . . . . 14  |-  F/ x [. y  /  x ]. ta
3025, 29nfxfr 1557 . . . . . . . . . . . . 13  |-  F/ x ta'
3130nfex 1767 . . . . . . . . . . . 12  |-  F/ x E. f ta'
3228, 31nfan 1771 . . . . . . . . . . 11  |-  F/ x
( R  FrSe  A  /\  E. f ta' )
3330nfeu 2159 . . . . . . . . . . 11  |-  F/ x E! f ta'
3432, 33nfim 1769 . . . . . . . . . 10  |-  F/ x
( ( R  FrSe  A  /\  E. f ta' )  ->  E! f ta' )
35 sneq 3651 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  { x }  =  { y } )
36 bnj1318 29055 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
3735, 36uneq12d 3330 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
3837eqeq2d 2294 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
3938anbi2d 684 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <-> 
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
4021, 22, 23, 24, 25bnj1373 29060 . . . . . . . . . . . . . 14  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
4139, 40syl6bbr 254 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <->  ta' ) )
4241exbidv 1612 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) )  <->  E. f ta' ) )
4342anbi2d 684 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( R  FrSe  A  /\  E. f ( f  e.  C  /\  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) ) ) )  <->  ( R  FrSe  A  /\  E. f ta' ) ) )
4441eubidv 2151 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( E! f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) )  <->  E! f ta' ) )
4543, 44imbi12d 311 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( ( R  FrSe  A  /\  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )  ->  E! f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )  <->  ( ( R 
FrSe  A  /\  E. f ta' )  ->  E! f ta' ) ) )
46 biid 227 . . . . . . . . . . 11  |-  ( ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <-> 
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
4721, 22, 23, 46bnj1321 29057 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) )  ->  E! f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
4834, 45, 47chvar 1926 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  E. f ta' )  ->  E! f ta' )
4920, 27, 48syl2anc 642 . . . . . . . 8  |-  ( ( ch  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  E! f ta' )
5049ex 423 . . . . . . 7  |-  ( ch 
->  ( y  e.  pred ( x ,  A ,  R )  ->  E! f ta' ) )
5117, 50ralrimi 2624 . . . . . 6  |-  ( ch 
->  A. y  e.  pred  ( x ,  A ,  R ) E! f ta' )
52 bnj1489.9 . . . . . . 7  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
5352a1i 10 . . . . . 6  |-  ( ch 
->  H  =  {
f  |  E. y  e.  pred  ( x ,  A ,  R ) ta' } )
54 biid 227 . . . . . . 7  |-  ( ( 
pred ( x ,  A ,  R )  e.  _V  /\  A. y  e.  pred  ( x ,  A ,  R
) E! f ta'  /\  H  =  { f  |  E. y  e. 
pred  ( x ,  A ,  R ) ta' } )  <->  (  pred ( x ,  A ,  R )  e.  _V  /\ 
A. y  e.  pred  ( x ,  A ,  R ) E! f ta'  /\  H  =  {
f  |  E. y  e.  pred  ( x ,  A ,  R ) ta' } ) )
5554bnj1366 28862 . . . . . 6  |-  ( ( 
pred ( x ,  A ,  R )  e.  _V  /\  A. y  e.  pred  ( x ,  A ,  R
) E! f ta'  /\  H  =  { f  |  E. y  e. 
pred  ( x ,  A ,  R ) ta' } )  ->  H  e.  _V )
5612, 51, 53, 55syl3anc 1182 . . . . 5  |-  ( ch 
->  H  e.  _V )
57 uniexg 4517 . . . . 5  |-  ( H  e.  _V  ->  U. H  e.  _V )
5856, 57syl 15 . . . 4  |-  ( ch 
->  U. H  e.  _V )
592, 58syl5eqel 2367 . . 3  |-  ( ch 
->  P  e.  _V )
60 snex 4216 . . . 4  |-  { <. x ,  ( G `  Z ) >. }  e.  _V
6160a1i 10 . . 3  |-  ( ch 
->  { <. x ,  ( G `  Z )
>. }  e.  _V )
6259, 61bnj1149 28824 . 2  |-  ( ch 
->  ( P  u.  { <. x ,  ( G `
 Z ) >. } )  e.  _V )
631, 62syl5eqel 2367 1  |-  ( ch 
->  Q  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788   [.wsbc 2991    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    Se w-bnj13 28715    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1312  29088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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