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Theorem bnj1312 29364
Description: Technical lemma for bnj60 29368. 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
bnj1312.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1312.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1312.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1312.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1312.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1312.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1312.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1312.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1312.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1312.10  |-  P  = 
U. H
bnj1312.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1312.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1312.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1312.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj1312  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Distinct variable groups:    A, d,
f, x, y, z    B, f    y, C    y, D    E, d, f, y, z    G, d, f, x, y, z    z, Q    R, d, f, x, y, z    z, Y    ch, z    ps, y    ta, y
Allowed substitution hints:    ps( x, z, f, d)    ch( x, y, f, d)    ta( x, z, f, d)    B( x, y, z, d)    C( x, z, f, d)    D( x, z, f, d)    P( x, y, z, f, d)    Q( x, y, f, d)    E( x)    H( x, y, z, f, d)    W( x, y, z, f, d)    Y( x, y, f, d)    Z( x, y, z, f, d)    ta'( x, y, z, f, d)

Proof of Theorem bnj1312
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1312.5 . . 3  |-  D  =  { x  e.  A  |  -.  E. f ta }
2 bnj1312.6 . . . 4  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
32simplbi 447 . . . . . . 7  |-  ( ps 
->  R  FrSe  A )
41bnj21 29019 . . . . . . . 8  |-  D  C_  A
54a1i 11 . . . . . . 7  |-  ( ps 
->  D  C_  A )
62simprbi 451 . . . . . . 7  |-  ( ps 
->  D  =/=  (/) )
71bnj1230 29111 . . . . . . . 8  |-  ( w  e.  D  ->  A. x  w  e.  D )
87bnj1228 29317 . . . . . . 7  |-  ( ( R  FrSe  A  /\  D  C_  A  /\  D  =/=  (/) )  ->  E. x  e.  D  A. y  e.  D  -.  y R x )
93, 5, 6, 8syl3anc 1184 . . . . . 6  |-  ( ps 
->  E. x  e.  D  A. y  e.  D  -.  y R x )
10 bnj1312.7 . . . . . 6  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
11 nfv 1629 . . . . . . . . 9  |-  F/ x  R  FrSe  A
127nfcii 2562 . . . . . . . . . 10  |-  F/_ x D
13 nfcv 2571 . . . . . . . . . 10  |-  F/_ x (/)
1412, 13nfne 2689 . . . . . . . . 9  |-  F/ x  D  =/=  (/)
1511, 14nfan 1846 . . . . . . . 8  |-  F/ x
( R  FrSe  A  /\  D  =/=  (/) )
162, 15nfxfr 1579 . . . . . . 7  |-  F/ x ps
1716nfri 1778 . . . . . 6  |-  ( ps 
->  A. x ps )
189, 10, 17bnj1521 29159 . . . . 5  |-  ( ps 
->  E. x ch )
1910simp2bi 973 . . . . 5  |-  ( ch 
->  x  e.  D
)
201bnj1538 29163 . . . . . 6  |-  ( x  e.  D  ->  -.  E. f ta )
21 bnj1312.1 . . . . . . . . 9  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
22 bnj1312.2 . . . . . . . . 9  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
23 bnj1312.3 . . . . . . . . 9  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
24 bnj1312.4 . . . . . . . . 9  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
25 bnj1312.8 . . . . . . . . 9  |-  ( ta'  <->  [. y  /  x ]. ta )
26 bnj1312.9 . . . . . . . . 9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
27 bnj1312.10 . . . . . . . . 9  |-  P  = 
U. H
28 bnj1312.11 . . . . . . . . 9  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
29 bnj1312.12 . . . . . . . . 9  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
3021, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29bnj1489 29362 . . . . . . . 8  |-  ( ch 
->  Q  e.  _V )
31 bnj1312.13 . . . . . . . . . . 11  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
32 bnj1312.14 . . . . . . . . . . 11  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
3310, 3bnj835 29065 . . . . . . . . . . . . . 14  |-  ( ch 
->  R  FrSe  A )
3421, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1384 29338 . . . . . . . . . . . . . 14  |-  ( R 
FrSe  A  ->  Fun  P
)
3533, 34syl 16 . . . . . . . . . . . . 13  |-  ( ch 
->  Fun  P )
3621, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1415 29344 . . . . . . . . . . . . 13  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
3735, 36bnj1422 29146 . . . . . . . . . . . 12  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
3821, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36bnj1416 29345 . . . . . . . . . . . . . 14  |-  ( ch 
->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
3921, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36bnj1421 29348 . . . . . . . . . . . . 13  |-  ( ch 
->  Fun  Q )
4039, 38bnj1422 29146 . . . . . . . . . . . 12  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
4121, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40bnj1423 29357 . . . . . . . . . . 11  |-  ( ch 
->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
4232fneq2i 5532 . . . . . . . . . . . 12  |-  ( Q  Fn  E  <->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
4340, 42sylibr 204 . . . . . . . . . . 11  |-  ( ch 
->  Q  Fn  E
)
4421, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32bnj1452 29358 . . . . . . . . . . 11  |-  ( ch 
->  E  e.  B
)
4521, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44bnj1463 29361 . . . . . . . . . 10  |-  ( ch 
->  Q  e.  C
)
4645, 38jca 519 . . . . . . . . 9  |-  ( ch 
->  ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
4721, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46bnj1491 29363 . . . . . . . 8  |-  ( ( ch  /\  Q  e. 
_V )  ->  E. f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
4830, 47mpdan 650 . . . . . . 7  |-  ( ch 
->  E. f ( f  e.  C  /\  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) ) ) )
4948, 24bnj1198 29104 . . . . . 6  |-  ( ch 
->  E. f ta )
5020, 49nsyl3 113 . . . . 5  |-  ( ch 
->  -.  x  e.  D
)
5118, 19, 50bnj1304 29128 . . . 4  |-  -.  ps
522, 51bnj1541 29164 . . 3  |-  ( R 
FrSe  A  ->  D  =  (/) )
531, 52bnj1476 29155 . 2  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f ta )
5424exbii 1592 . . . 4  |-  ( E. f ta  <->  E. f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
55 df-rex 2703 . . . 4  |-  ( E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  <->  E. f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
5654, 55bitr4i 244 . . 3  |-  ( E. f ta  <->  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
5756ralbii 2721 . 2  |-  ( A. x  e.  A  E. f ta  <->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
5853, 57sylib 189 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948   [.wsbc 3153    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   U.cuni 4007   class class class wbr 4204   dom cdm 4870    |` cres 4872   Fun wfun 5440    Fn wfn 5441   ` cfv 5446    predc-bnj14 28989    FrSe w-bnj15 28993    trClc-bnj18 28995
This theorem is referenced by:  bnj1493  29365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-bnj17 28988  df-bnj14 28990  df-bnj13 28992  df-bnj15 28994  df-bnj18 28996  df-bnj19 28998
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