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Theorem bnj1489 29452
Description: Technical lemma for bnj60 29458. 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 29424 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  R  Se  A )
6 df-bnj13 29083 . . . . . . . . . 10  |-  ( R  Se  A  <->  A. x  e.  A  pred ( x ,  A ,  R
)  e.  _V )
75, 6sylib 198 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  A. x  e.  A  pred ( x ,  A ,  R
)  e.  _V )
84, 7bnj832 29155 . . . . . . . 8  |-  ( ps 
->  A. x  e.  A  pred ( x ,  A ,  R )  e.  _V )
93, 8bnj835 29157 . . . . . . 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 29198 . . . . . . 7  |-  ( ch 
->  x  e.  A
)
129, 11bnj1294 29216 . . . . . 6  |-  ( ch 
->  pred ( x ,  A ,  R )  e.  _V )
13 nfv 1730 . . . . . . . . 9  |-  F/ y ps
14 nfv 1730 . . . . . . . . 9  |-  F/ y  x  e.  D
15 nfra1 2787 . . . . . . . . 9  |-  F/ y A. y  e.  D  -.  y R x
1613, 14, 15nf3an 1960 . . . . . . . 8  |-  F/ y ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
173, 16nfxfr 1668 . . . . . . 7  |-  F/ y ch
184simplbi 460 . . . . . . . . . . 11  |-  ( ps 
->  R  FrSe  A )
193, 18bnj835 29157 . . . . . . . . . 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 29429 . . . . . . . . . 10  |-  ( ch 
->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
2726r19.21bi 2775 . . . . . . . . 9  |-  ( ( ch  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  E. f ta' )
28 nfv 1730 . . . . . . . . . . . 12  |-  F/ x  R  FrSe  A
29 nfsbc1v 3299 . . . . . . . . . . . . . 14  |-  F/ x [. y  /  x ]. ta
3025, 29nfxfr 1668 . . . . . . . . . . . . 13  |-  F/ x ta'
3130nfex 1978 . . . . . . . . . . . 12  |-  F/ x E. f ta'
3228, 31nfan 1958 . . . . . . . . . . 11  |-  F/ x
( R  FrSe  A  /\  E. f ta' )
3330nfeu 2258 . . . . . . . . . . 11  |-  F/ x E! f ta'
3432, 33nfim 1950 . . . . . . . . . 10  |-  F/ x
( ( R  FrSe  A  /\  E. f ta' )  ->  E! f ta' )
35 sneq 3984 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  { x }  =  { y } )
36 bnj1318 29421 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
3735, 36uneq12d 3600 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
3837eqeq2d 2418 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
3938anbi2d 704 . . . . . . . . . . . . . 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 29426 . . . . . . . . . . . . . 14  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
4139, 40syl6bbr 265 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <->  ta' ) )
4241exbidv 1737 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) )  <->  E. f ta' ) )
4342anbi2d 704 . . . . . . . . . . 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 2262 . . . . . . . . . . 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 238 . . . . . . . . . . 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 29423 . . . . . . . . . 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 2042 . . . . . . . . 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 2806 . . . . . 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 238 . . . . . . 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 29228 . . . . . 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 1232 . . . . 5  |-  ( ch 
->  H  e.  _V )
57 uniexg 6581 . . . . 5  |-  ( H  e.  _V  ->  U. H  e.  _V )
5856, 57syl 17 . . . 4  |-  ( ch 
->  U. H  e.  _V )
592, 58syl5eqel 2496 . . 3  |-  ( ch 
->  P  e.  _V )
60 snex 4634 . . . 4  |-  { <. x ,  ( G `  Z ) >. }  e.  _V
6160a1i 11 . . 3  |-  ( ch 
->  { <. x ,  ( G `  Z )
>. }  e.  _V )
6259, 61bnj1149 29191 . 2  |-  ( ch 
->  ( P  u.  { <. x ,  ( G `
 Z ) >. } )  e.  _V )
631, 62syl5eqel 2496 1  |-  ( ch 
->  Q  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407   E.wex 1635    e. wcel 1844   E!weu 2240   {cab 2389    =/= wne 2600   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3061   [.wsbc 3279    u. cun 3414    C_ wss 3416   (/)c0 3740   {csn 3974   <.cop 3980   U.cuni 4193   class class class wbr 4397   dom cdm 4825    |` cres 4827    Fn wfn 5566   ` cfv 5571    predc-bnj14 29080    Se w-bnj13 29082    FrSe w-bnj15 29084    trClc-bnj18 29086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-reg 8054  ax-inf2 8093
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-om 6686  df-1o 7169  df-bnj17 29079  df-bnj14 29081  df-bnj13 29083  df-bnj15 29085  df-bnj18 29087  df-bnj19 29089
This theorem is referenced by:  bnj1312  29454
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