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Theorem bnj1489 33209
Description: Technical lemma for bnj60 33215. 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 33181 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  R  Se  A )
6 df-bnj13 32841 . . . . . . . . . 10  |-  ( R  Se  A  <->  A. x  e.  A  pred ( x ,  A ,  R
)  e.  _V )
75, 6sylib 196 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  A. x  e.  A  pred ( x ,  A ,  R
)  e.  _V )
84, 7bnj832 32912 . . . . . . . 8  |-  ( ps 
->  A. x  e.  A  pred ( x ,  A ,  R )  e.  _V )
93, 8bnj835 32914 . . . . . . 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 32955 . . . . . . 7  |-  ( ch 
->  x  e.  A
)
129, 11bnj1294 32973 . . . . . 6  |-  ( ch 
->  pred ( x ,  A ,  R )  e.  _V )
13 nfv 1683 . . . . . . . . 9  |-  F/ y ps
14 nfv 1683 . . . . . . . . 9  |-  F/ y  x  e.  D
15 nfra1 2845 . . . . . . . . 9  |-  F/ y A. y  e.  D  -.  y R x
1613, 14, 15nf3an 1877 . . . . . . . 8  |-  F/ y ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
173, 16nfxfr 1625 . . . . . . 7  |-  F/ y ch
184simplbi 460 . . . . . . . . . . 11  |-  ( ps 
->  R  FrSe  A )
193, 18bnj835 32914 . . . . . . . . . 10  |-  ( ch 
->  R  FrSe  A )
2019adantr 465 . . . . . . . . 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 33186 . . . . . . . . . 10  |-  ( ch 
->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
2726r19.21bi 2833 . . . . . . . . 9  |-  ( ( ch  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  E. f ta' )
28 nfv 1683 . . . . . . . . . . . 12  |-  F/ x  R  FrSe  A
29 nfsbc1v 3351 . . . . . . . . . . . . . 14  |-  F/ x [. y  /  x ]. ta
3025, 29nfxfr 1625 . . . . . . . . . . . . 13  |-  F/ x ta'
3130nfex 1895 . . . . . . . . . . . 12  |-  F/ x E. f ta'
3228, 31nfan 1875 . . . . . . . . . . 11  |-  F/ x
( R  FrSe  A  /\  E. f ta' )
3330nfeu 2294 . . . . . . . . . . 11  |-  F/ x E! f ta'
3432, 33nfim 1867 . . . . . . . . . 10  |-  F/ x
( ( R  FrSe  A  /\  E. f ta' )  ->  E! f ta' )
35 sneq 4037 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  { x }  =  { y } )
36 bnj1318 33178 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
3735, 36uneq12d 3659 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
3837eqeq2d 2481 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
3938anbi2d 703 . . . . . . . . . . . . . 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 33183 . . . . . . . . . . . . . 14  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
4139, 40syl6bbr 263 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <->  ta' ) )
4241exbidv 1690 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) )  <->  E. f ta' ) )
4342anbi2d 703 . . . . . . . . . . 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 2298 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( E! f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) )  <->  E! f ta' ) )
4543, 44imbi12d 320 . . . . . . . . . 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 236 . . . . . . . . . . 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 33180 . . . . . . . . . 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 1982 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  E. f ta' )  ->  E! f ta' )
4920, 27, 48syl2anc 661 . . . . . . . 8  |-  ( ( ch  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  E! f ta' )
5049ex 434 . . . . . . 7  |-  ( ch 
->  ( y  e.  pred ( x ,  A ,  R )  ->  E! f ta' ) )
5117, 50ralrimi 2864 . . . . . 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 11 . . . . . 6  |-  ( ch 
->  H  =  {
f  |  E. y  e.  pred  ( x ,  A ,  R ) ta' } )
54 biid 236 . . . . . . 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 32985 . . . . . 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 1228 . . . . 5  |-  ( ch 
->  H  e.  _V )
57 uniexg 6581 . . . . 5  |-  ( H  e.  _V  ->  U. H  e.  _V )
5856, 57syl 16 . . . 4  |-  ( ch 
->  U. H  e.  _V )
592, 58syl5eqel 2559 . . 3  |-  ( ch 
->  P  e.  _V )
60 snex 4688 . . . 4  |-  { <. x ,  ( G `  Z ) >. }  e.  _V
6160a1i 11 . . 3  |-  ( ch 
->  { <. x ,  ( G `  Z )
>. }  e.  _V )
6259, 61bnj1149 32948 . 2  |-  ( ch 
->  ( P  u.  { <. x ,  ( G `
 Z ) >. } )  e.  _V )
631, 62syl5eqel 2559 1  |-  ( ch 
->  Q  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   [.wsbc 3331    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588    predc-bnj14 32838    Se w-bnj13 32840    FrSe w-bnj15 32842    trClc-bnj18 32844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-bnj17 32837  df-bnj14 32839  df-bnj13 32841  df-bnj15 32843  df-bnj18 32845  df-bnj19 32847
This theorem is referenced by:  bnj1312  33211
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