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Theorem bnj1415 33962
Description: Technical lemma for bnj60 33986. 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 33685 . . 3  |-  ( ch 
->  R  FrSe  A )
5 bnj1415.5 . . . 4  |-  D  =  { x  e.  A  |  -.  E. f ta }
65, 1bnj1212 33726 . . 3  |-  ( ch 
->  x  e.  A
)
7 eqid 2443 . . . 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 33961 . . 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 4396 . . . 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 4370 . . . . 5  |-  U_ y  e.  pred  ( x ,  A ,  R ) { y }  =  pred ( x ,  A ,  R )
1211uneq1i 3639 . . . 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 2472 . . 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 33958 . . 3  |-  ( ch 
->  U_ y  e.  pred  ( x ,  A ,  R ) ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
2413, 23syl5eqr 2498 . 2  |-  ( ch 
->  (  pred ( x ,  A ,  R
)  u.  U_ y  e.  pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )  =  dom  P
)
259, 24eqtr2d 2485 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 974    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428    =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797   [.wsbc 3313    u. cun 3459    C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020   U.cuni 4234   U_ciun 4315   class class class wbr 4437   dom cdm 4989    |` cres 4991    Fn wfn 5573   ` cfv 5578    predc-bnj14 33608    FrSe w-bnj15 33612    trClc-bnj18 33614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-bnj17 33607  df-bnj14 33609  df-bnj13 33611  df-bnj15 33613  df-bnj18 33615  df-bnj19 33617
This theorem is referenced by:  bnj1312  33982
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