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Theorem bnj1423 28751
Description: Technical lemma for bnj60 28762. 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 28749 . . 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 2380 . . . 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 28750 . . 3  |-  ( ( ( ch  /\  z  e.  E )  /\  z  e.  trCl ( x ,  A ,  R ) )  ->  ( Q `  z )  =  ( G `  W ) )
2614bnj1424 28541 . . . 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 2725 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 1547    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   A.wral 2642   E.wrex 2643   {crab 2646   [.wsbc 3097    u. cun 3254    C_ wss 3256   (/)c0 3564   {csn 3750   <.cop 3753   U.cuni 3950   class class class wbr 4146   dom cdm 4811    |` cres 4813    Fn wfn 5382   ` cfv 5387    /\ w-bnj17 28381    predc-bnj14 28383    FrSe w-bnj15 28387    trClc-bnj18 28389
This theorem is referenced by:  bnj1312  28758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-reg 7486  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-1o 6653  df-bnj17 28382  df-bnj14 28384  df-bnj13 28386  df-bnj15 28388  df-bnj18 28390
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