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Theorem bnj1463 32069
Description: Technical lemma for bnj60 32076. 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
bnj1463.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1463.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1463.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1463.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1463.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1463.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1463.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1463.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1463.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1463.10  |-  P  = 
U. H
bnj1463.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1463.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1463.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1463.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1463.15  |-  ( ch 
->  Q  e.  _V )
bnj1463.16  |-  ( ch 
->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
bnj1463.17  |-  ( ch 
->  Q  Fn  E
)
bnj1463.18  |-  ( ch 
->  E  e.  B
)
Assertion
Ref Expression
bnj1463  |-  ( ch 
->  Q  e.  C
)
Distinct variable groups:    A, d,
f, x    B, f    E, d, z    G, d, f, x, z    z, Q    R, d, f, x   
z, Y    y, d, x
Allowed substitution hints:    ps( x, y, z, f, d)    ch( x, y, z, f, d)    ta( x, y, z, f, d)    A( y, z)    B( x, y, z, d)    C( x, y, z, f, d)    D( x, y, z, f, d)    P( x, y, z, f, d)    Q( x, y, f, d)    R( y, z)    E( x, y, f)    G( y)    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 bnj1463
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1463.18 . . . . . . 7  |-  ( ch 
->  E  e.  B
)
2 elex 3000 . . . . . . 7  |-  ( E  e.  B  ->  E  e.  _V )
31, 2syl 16 . . . . . 6  |-  ( ch 
->  E  e.  _V )
4 eleq1 2503 . . . . . . . 8  |-  ( d  =  E  ->  (
d  e.  B  <->  E  e.  B ) )
5 fneq2 5519 . . . . . . . . 9  |-  ( d  =  E  ->  ( Q  Fn  d  <->  Q  Fn  E ) )
6 raleq 2936 . . . . . . . . 9  |-  ( d  =  E  ->  ( A. z  e.  d 
( Q `  z
)  =  ( G `
 W )  <->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) )
75, 6anbi12d 710 . . . . . . . 8  |-  ( d  =  E  ->  (
( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) )  <-> 
( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) ) )
84, 7anbi12d 710 . . . . . . 7  |-  ( d  =  E  ->  (
( d  e.  B  /\  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) )  <->  ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) ) ) )
9 bnj1463.1 . . . . . . . . . . . 12  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
109bnj1317 31838 . . . . . . . . . . 11  |-  ( w  e.  B  ->  A. d  w  e.  B )
1110nfcii 2579 . . . . . . . . . 10  |-  F/_ d B
1211nfel2 2606 . . . . . . . . 9  |-  F/ d  E  e.  B
13 bnj1463.2 . . . . . . . . . . . . 13  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
14 bnj1463.3 . . . . . . . . . . . . 13  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
15 bnj1463.4 . . . . . . . . . . . . 13  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
16 bnj1463.5 . . . . . . . . . . . . 13  |-  D  =  { x  e.  A  |  -.  E. f ta }
17 bnj1463.6 . . . . . . . . . . . . 13  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
18 bnj1463.7 . . . . . . . . . . . . 13  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
19 bnj1463.8 . . . . . . . . . . . . 13  |-  ( ta'  <->  [. y  /  x ]. ta )
20 bnj1463.9 . . . . . . . . . . . . 13  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
21 bnj1463.10 . . . . . . . . . . . . 13  |-  P  = 
U. H
22 bnj1463.11 . . . . . . . . . . . . 13  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
23 bnj1463.12 . . . . . . . . . . . . 13  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
249, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23bnj1467 32068 . . . . . . . . . . . 12  |-  ( w  e.  Q  ->  A. d  w  e.  Q )
2524nfcii 2579 . . . . . . . . . . 11  |-  F/_ d Q
26 nfcv 2589 . . . . . . . . . . 11  |-  F/_ d E
2725, 26nffn 5526 . . . . . . . . . 10  |-  F/ d  Q  Fn  E
28 bnj1463.13 . . . . . . . . . . . . 13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
299, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28bnj1446 32059 . . . . . . . . . . . 12  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
3029nfi 1596 . . . . . . . . . . 11  |-  F/ d ( Q `  z
)  =  ( G `
 W )
3126, 30nfral 2788 . . . . . . . . . 10  |-  F/ d A. z  e.  E  ( Q `  z )  =  ( G `  W )
3227, 31nfan 1861 . . . . . . . . 9  |-  F/ d ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
3312, 32nfan 1861 . . . . . . . 8  |-  F/ d ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) )
3433nfri 1808 . . . . . . 7  |-  ( ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) )  ->  A. d ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) ) )
35 bnj1463.17 . . . . . . . 8  |-  ( ch 
->  Q  Fn  E
)
36 bnj1463.16 . . . . . . . 8  |-  ( ch 
->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
371, 35, 36jca32 535 . . . . . . 7  |-  ( ch 
->  ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) ) )
388, 34, 37bnj1465 31861 . . . . . 6  |-  ( ( ch  /\  E  e. 
_V )  ->  E. d
( d  e.  B  /\  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) ) )
393, 38mpdan 668 . . . . 5  |-  ( ch 
->  E. d ( d  e.  B  /\  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) ) ) )
40 df-rex 2740 . . . . 5  |-  ( E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) )  <->  E. d
( d  e.  B  /\  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) ) )
4139, 40sylibr 212 . . . 4  |-  ( ch 
->  E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) )
42 bnj1463.15 . . . . 5  |-  ( ch 
->  Q  e.  _V )
43 nfcv 2589 . . . . . . . 8  |-  F/_ f B
449, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23bnj1466 32067 . . . . . . . . . . 11  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
4544nfcii 2579 . . . . . . . . . 10  |-  F/_ f Q
46 nfcv 2589 . . . . . . . . . 10  |-  F/_ f
d
4745, 46nffn 5526 . . . . . . . . 9  |-  F/ f  Q  Fn  d
489, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28bnj1448 32061 . . . . . . . . . . 11  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. f
( Q `  z
)  =  ( G `
 W ) )
4948nfi 1596 . . . . . . . . . 10  |-  F/ f ( Q `  z
)  =  ( G `
 W )
5046, 49nfral 2788 . . . . . . . . 9  |-  F/ f A. z  e.  d  ( Q `  z
)  =  ( G `
 W )
5147, 50nfan 1861 . . . . . . . 8  |-  F/ f ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) )
5243, 51nfrex 2790 . . . . . . 7  |-  F/ f E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) )
5352nfri 1808 . . . . . 6  |-  ( E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) )  ->  A. f E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) ) )
5425nfeq2 2605 . . . . . . 7  |-  F/ d  f  =  Q
55 fneq1 5518 . . . . . . . 8  |-  ( f  =  Q  ->  (
f  Fn  d  <->  Q  Fn  d ) )
56 fveq1 5709 . . . . . . . . . 10  |-  ( f  =  Q  ->  (
f `  z )  =  ( Q `  z ) )
57 reseq1 5123 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (
f  |`  pred ( z ,  A ,  R ) )  =  ( Q  |`  pred ( z ,  A ,  R ) ) )
5857opeq2d 4085 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.  =  <. z ,  ( Q  |`  pred ( z ,  A ,  R
) ) >. )
5958, 28syl6eqr 2493 . . . . . . . . . . 11  |-  ( f  =  Q  ->  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.  =  W )
6059fveq2d 5714 . . . . . . . . . 10  |-  ( f  =  Q  ->  ( G `  <. z ,  ( f  |`  pred (
z ,  A ,  R ) ) >.
)  =  ( G `
 W ) )
6156, 60eqeq12d 2457 . . . . . . . . 9  |-  ( f  =  Q  ->  (
( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )  <->  ( Q `  z )  =  ( G `  W ) ) )
6261ralbidv 2754 . . . . . . . 8  |-  ( f  =  Q  ->  ( A. z  e.  d 
( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )  <->  A. z  e.  d  ( Q `  z )  =  ( G `  W ) ) )
6355, 62anbi12d 710 . . . . . . 7  |-  ( f  =  Q  ->  (
( f  Fn  d  /\  A. z  e.  d  ( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
)  <->  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) ) ) )
6454, 63rexbid 2753 . . . . . 6  |-  ( f  =  Q  ->  ( E. d  e.  B  ( f  Fn  d  /\  A. z  e.  d  ( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
)  <->  E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) ) )
6553, 64, 44bnj1468 31862 . . . . 5  |-  ( Q  e.  _V  ->  ( [. Q  /  f ]. E. d  e.  B  ( f  Fn  d  /\  A. z  e.  d  ( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
)  <->  E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) ) )
6642, 65syl 16 . . . 4  |-  ( ch 
->  ( [. Q  / 
f ]. E. d  e.  B  ( f  Fn  d  /\  A. z  e.  d  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) )  <->  E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) ) ) )
6741, 66mpbird 232 . . 3  |-  ( ch 
->  [. Q  /  f ]. E. d  e.  B  ( f  Fn  d  /\  A. z  e.  d  ( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
) )
68 fveq2 5710 . . . . . . . 8  |-  ( x  =  z  ->  (
f `  x )  =  ( f `  z ) )
69 id 22 . . . . . . . . . . 11  |-  ( x  =  z  ->  x  =  z )
70 bnj602 31931 . . . . . . . . . . . 12  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
7170reseq2d 5129 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
f  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( z ,  A ,  R ) ) )
7269, 71opeq12d 4086 . . . . . . . . . 10  |-  ( x  =  z  ->  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
7313, 72syl5eq 2487 . . . . . . . . 9  |-  ( x  =  z  ->  Y  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
7473fveq2d 5714 . . . . . . . 8  |-  ( x  =  z  ->  ( G `  Y )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >. ) )
7568, 74eqeq12d 2457 . . . . . . 7  |-  ( x  =  z  ->  (
( f `  x
)  =  ( G `
 Y )  <->  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) ) )
7675cbvralv 2966 . . . . . 6  |-  ( A. x  e.  d  (
f `  x )  =  ( G `  Y )  <->  A. z  e.  d  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) )
7776anbi2i 694 . . . . 5  |-  ( ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  ( f  Fn  d  /\  A. z  e.  d  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) ) )
7877rexbii 2759 . . . 4  |-  ( E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  E. d  e.  B  ( f  Fn  d  /\  A. z  e.  d  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) ) )
7978sbcbii 3265 . . 3  |-  ( [. Q  /  f ]. E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  [. Q  / 
f ]. E. d  e.  B  ( f  Fn  d  /\  A. z  e.  d  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) ) )
8067, 79sylibr 212 . 2  |-  ( ch 
->  [. Q  /  f ]. E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
8114bnj1454 31858 . . 3  |-  ( Q  e.  _V  ->  ( Q  e.  C  <->  [. Q  / 
f ]. E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
8242, 81syl 16 . 2  |-  ( ch 
->  ( Q  e.  C  <->  [. Q  /  f ]. E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) ) )
8380, 82mpbird 232 1  |-  ( ch 
->  Q  e.  C
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2620   A.wral 2734   E.wrex 2735   {crab 2738   _Vcvv 2991   [.wsbc 3205    u. cun 3345    C_ wss 3347   (/)c0 3656   {csn 3896   <.cop 3902   U.cuni 4110   class class class wbr 4311   dom cdm 4859    |` cres 4861    Fn wfn 5432   ` cfv 5437    predc-bnj14 31699    FrSe w-bnj15 31703    trClc-bnj18 31705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-res 4871  df-iota 5400  df-fun 5439  df-fn 5440  df-fv 5445  df-bnj14 31700
This theorem is referenced by:  bnj1312  32072
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