Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1423 Structured version   Unicode version

Theorem bnj1423 29357
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
bnj1423.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1423.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1423.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1423.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1423.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1423.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1423.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1423.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1423.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1423.10  |-  P  = 
U. H
bnj1423.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1423.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1423.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1423.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1423.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1423.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Assertion
Ref Expression
bnj1423  |-  ( ch 
->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
Distinct variable groups:    A, d,
f, x, y, z    B, f    y, D    E, d, f, y    G, d, f, x, y, z    R, d, f, x, y, z    z, Y    ch, z    ps, y
Allowed substitution hints:    ps( x, z, f, d)    ch( x, y, f, d)    ta( x, y, z, f, d)    B( x, y, z, d)    C( x, y, z, f, d)    D( x, z, f, d)    P( x, y, z, f, d)    Q( x, y, z, f, d)    E( x, z)    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 bnj1423
StepHypRef Expression
1 bnj1423.1 . . . 4  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1423.2 . . . 4  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1423.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1423.4 . . . 4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
5 bnj1423.5 . . . 4  |-  D  =  { x  e.  A  |  -.  E. f ta }
6 bnj1423.6 . . . 4  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
7 bnj1423.7 . . . 4  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
8 bnj1423.8 . . . 4  |-  ( ta'  <->  [. y  /  x ]. ta )
9 bnj1423.9 . . . 4  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
10 bnj1423.10 . . . 4  |-  P  = 
U. H
11 bnj1423.11 . . . 4  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
12 bnj1423.12 . . . 4  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
13 bnj1423.13 . . . 4  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
14 bnj1423.14 . . . 4  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
15 bnj1423.15 . . . 4  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
16 bnj1423.16 . . . 4  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
17 biid 228 . . . 4  |-  ( ( ch  /\  z  e.  E )  <->  ( ch  /\  z  e.  E ) )
18 biid 228 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  <-> 
( ( ch  /\  z  e.  E )  /\  z  e.  { x } ) )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18bnj1442 29355 . . 3  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  { x } )  ->  ( Q `  z )  =  ( G `  W ) )
20 biid 228 . . . 4  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  <->  ( ( ch 
/\  z  e.  E
)  /\  z  e.  trCl ( x ,  A ,  R ) ) )
21 biid 228 . . . 4  |-  ( ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  /\  f  e.  H  /\  z  e.  dom  f )  <-> 
( ( ( ch 
/\  z  e.  E
)  /\  z  e.  trCl ( x ,  A ,  R ) )  /\  f  e.  H  /\  z  e.  dom  f ) )
22 biid 228 . . . 4  |-  ( ( ( ( ( ch 
/\  z  e.  E
)  /\  z  e.  trCl ( x ,  A ,  R ) )  /\  f  e.  H  /\  z  e.  dom  f )  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) )  <-> 
( ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  /\  f  e.  H  /\  z  e. 
dom  f )  /\  y  e.  pred ( x ,  A ,  R
)  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
23 biid 228 . . . 4  |-  ( ( ( ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  /\  f  e.  H  /\  z  e. 
dom  f )  /\  y  e.  pred ( x ,  A ,  R
)  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  ( ( ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  /\  f  e.  H  /\  z  e.  dom  f )  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) )  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
24 eqid 2435 . . . 4  |-  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >.
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24bnj1450 29356 . . 3  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  ( Q `  z )  =  ( G `  W ) )
2614bnj1424 29147 . . . 4  |-  ( z  e.  E  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
2726adantl 453 . . 3  |-  ( ( ch  /\  z  e.  E )  ->  (
z  e.  { x }  \/  z  e.  trCl ( x ,  A ,  R ) ) )
2819, 25, 27mpjaodan 762 . 2  |-  ( ( ch  /\  z  e.  E )  ->  ( Q `  z )  =  ( G `  W ) )
2928ralrimiva 2781 1  |-  ( ch 
->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ 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   class class class wbr 4204   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446    /\ w-bnj17 28987    predc-bnj14 28989    FrSe w-bnj15 28993    trClc-bnj18 28995
This theorem is referenced by:  bnj1312  29364
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
  Copyright terms: Public domain W3C validator