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Theorem bnj1415 29308
Description: Technical lemma for bnj60 29332. 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
bnj1415.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1415.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1415.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1415.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1415.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1415.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1415.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1415.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1415.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1415.10  |-  P  = 
U. H
Assertion
Ref Expression
bnj1415  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
Distinct variable groups:    A, f, x, y    B, f    y, C    y, D    R, f, x, y    f, d, x    ps, y    ta, y
Allowed substitution hints:    ps( x, f, d)    ch( x, y, f, d)    ta( x, f, d)    A( d)    B( x, y, d)    C( x, f, d)    D( x, f, d)    P( x, y, f, d)    R( d)    G( x, y, f, d)    H( x, y, f, d)    Y( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1415
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj1415.7 . . . 4  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
2 bnj1415.6 . . . . 5  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
32simplbi 447 . . . 4  |-  ( ps 
->  R  FrSe  A )
41, 3bnj835 29029 . . 3  |-  ( ch 
->  R  FrSe  A )
5 bnj1415.5 . . . 4  |-  D  =  { x  e.  A  |  -.  E. f ta }
65, 1bnj1212 29072 . . 3  |-  ( ch 
->  x  e.  A
)
7 eqid 2435 . . . 4  |-  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )
87bnj1414 29307 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  =  (  pred (
x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
94, 6, 8syl2anc 643 . 2  |-  ( ch 
->  trCl ( x ,  A ,  R )  =  (  pred (
x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
10 iunun 4163 . . . 4  |-  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) )  =  ( U_ y  e.  pred  ( x ,  A ,  R
) { y }  u.  U_ y  e. 
pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )
11 iunid 4138 . . . . 5  |-  U_ y  e.  pred  ( x ,  A ,  R ) { y }  =  pred ( x ,  A ,  R )
1211uneq1i 3489 . . . 4  |-  ( U_ y  e.  pred  ( x ,  A ,  R
) { y }  u.  U_ y  e. 
pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )
1310, 12eqtri 2455 . . 3  |-  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) )  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )
14 bnj1415.1 . . . 4  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
15 bnj1415.2 . . . 4  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
16 bnj1415.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
17 bnj1415.4 . . . 4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
18 bnj1415.8 . . . 4  |-  ( ta'  <->  [. y  /  x ]. ta )
19 bnj1415.9 . . . 4  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
20 bnj1415.10 . . . 4  |-  P  = 
U. H
21 biid 228 . . . 4  |-  ( ( ch  /\  z  e. 
U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) ) )  <-> 
( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
22 biid 228 . . . 4  |-  ( ( ( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) ) )  /\  y  e.  pred ( x ,  A ,  R )  /\  z  e.  ( { y }  u.  trCl ( y ,  A ,  R ) ) )  <->  ( ( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R
) ( { y }  u.  trCl (
y ,  A ,  R ) ) )  /\  y  e.  pred ( x ,  A ,  R )  /\  z  e.  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
2314, 15, 16, 17, 5, 2, 1, 18, 19, 20, 21, 22bnj1398 29304 . . 3  |-  ( ch 
->  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
2413, 23syl5eqr 2481 . 2  |-  ( ch 
->  (  pred ( x ,  A ,  R
)  u.  U_ y  e.  pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )  =  dom  P
)
259, 24eqtr2d 2468 1  |-  ( ch 
->  dom  P  =  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   [.wsbc 3153    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   U.cuni 4007   U_ciun 4085   class class class wbr 4204   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446    predc-bnj14 28953    FrSe w-bnj15 28957    trClc-bnj18 28959
This theorem is referenced by:  bnj1312  29328
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 7550  ax-inf2 7586
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 28952  df-bnj14 28954  df-bnj13 28956  df-bnj15 28958  df-bnj18 28960  df-bnj19 28962
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