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Theorem bnj1415 33191
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
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 460 . . . 4  |-  ( ps 
->  R  FrSe  A )
41, 3bnj835 32914 . . 3  |-  ( ch 
->  R  FrSe  A )
5 bnj1415.5 . . . 4  |-  D  =  { x  e.  A  |  -.  E. f ta }
65, 1bnj1212 32955 . . 3  |-  ( ch 
->  x  e.  A
)
7 eqid 2467 . . . 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 33190 . . 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 661 . 2  |-  ( ch 
->  trCl ( x ,  A ,  R )  =  (  pred (
x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
10 iunun 4406 . . . 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 4380 . . . . 5  |-  U_ y  e.  pred  ( x ,  A ,  R ) { y }  =  pred ( x ,  A ,  R )
1211uneq1i 3654 . . . 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 2496 . . 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 236 . . . 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 236 . . . 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 33187 . . 3  |-  ( ch 
->  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
2413, 23syl5eqr 2522 . 2  |-  ( ch 
->  (  pred ( x ,  A ,  R
)  u.  U_ y  e.  pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )  =  dom  P
)
259, 24eqtr2d 2509 1  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
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   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   [.wsbc 3331    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325   class class class wbr 4447   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588    predc-bnj14 32838    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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