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Theorem bnj1463 29361
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
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 2956 . . . . . . 7  |-  ( E  e.  B  ->  E  e.  _V )
31, 2syl 16 . . . . . 6  |-  ( ch 
->  E  e.  _V )
4 eleq1 2495 . . . . . . . 8  |-  ( d  =  E  ->  (
d  e.  B  <->  E  e.  B ) )
5 fneq2 5527 . . . . . . . . 9  |-  ( d  =  E  ->  ( Q  Fn  d  <->  Q  Fn  E ) )
6 raleq 2896 . . . . . . . . 9  |-  ( d  =  E  ->  ( A. z  e.  d 
( Q `  z
)  =  ( G `
 W )  <->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) )
75, 6anbi12d 692 . . . . . . . 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 692 . . . . . . 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 29130 . . . . . . . . . . 11  |-  ( w  e.  B  ->  A. d  w  e.  B )
1110nfcii 2562 . . . . . . . . . 10  |-  F/_ d B
1211nfel2 2583 . . . . . . . . 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 29360 . . . . . . . . . . . 12  |-  ( w  e.  Q  ->  A. d  w  e.  Q )
2524nfcii 2562 . . . . . . . . . . 11  |-  F/_ d Q
26 nfcv 2571 . . . . . . . . . . 11  |-  F/_ d E
2725, 26nffn 5533 . . . . . . . . . 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 29351 . . . . . . . . . . . 12  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. d
( Q `  z
)  =  ( G `
 W ) )
3029nfi 1560 . . . . . . . . . . 11  |-  F/ d ( Q `  z
)  =  ( G `
 W )
3126, 30nfral 2751 . . . . . . . . . 10  |-  F/ d A. z  e.  E  ( Q `  z )  =  ( G `  W )
3227, 31nfan 1846 . . . . . . . . 9  |-  F/ d ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
3312, 32nfan 1846 . . . . . . . 8  |-  F/ d ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) )
3433nfri 1778 . . . . . . 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 522 . . . . . . 7  |-  ( ch 
->  ( E  e.  B  /\  ( Q  Fn  E  /\  A. z  e.  E  ( Q `  z )  =  ( G `  W ) ) ) )
388, 34, 37bnj1465 29153 . . . . . 6  |-  ( ( ch  /\  E  e. 
_V )  ->  E. d
( d  e.  B  /\  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) ) ) )
393, 38mpdan 650 . . . . 5  |-  ( ch 
->  E. d ( d  e.  B  /\  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z )  =  ( G `  W ) ) ) )
40 df-rex 2703 . . . . 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 204 . . . 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 2571 . . . . . . . 8  |-  F/_ f B
449, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23bnj1466 29359 . . . . . . . . . . 11  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
4544nfcii 2562 . . . . . . . . . 10  |-  F/_ f Q
46 nfcv 2571 . . . . . . . . . 10  |-  F/_ f
d
4745, 46nffn 5533 . . . . . . . . 9  |-  F/ f  Q  Fn  d
489, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28bnj1448 29353 . . . . . . . . . . 11  |-  ( ( Q `  z )  =  ( G `  W )  ->  A. f
( Q `  z
)  =  ( G `
 W ) )
4948nfi 1560 . . . . . . . . . 10  |-  F/ f ( Q `  z
)  =  ( G `
 W )
5046, 49nfral 2751 . . . . . . . . 9  |-  F/ f A. z  e.  d  ( Q `  z
)  =  ( G `
 W )
5147, 50nfan 1846 . . . . . . . 8  |-  F/ f ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) )
5243, 51nfrex 2753 . . . . . . 7  |-  F/ f E. d  e.  B  ( Q  Fn  d  /\  A. z  e.  d  ( Q `  z
)  =  ( G `
 W ) )
5352nfri 1778 . . . . . 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 2582 . . . . . . 7  |-  F/ d  f  =  Q
55 fneq1 5526 . . . . . . . 8  |-  ( f  =  Q  ->  (
f  Fn  d  <->  Q  Fn  d ) )
56 fveq1 5719 . . . . . . . . . 10  |-  ( f  =  Q  ->  (
f `  z )  =  ( Q `  z ) )
57 reseq1 5132 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (
f  |`  pred ( z ,  A ,  R ) )  =  ( Q  |`  pred ( z ,  A ,  R ) ) )
5857opeq2d 3983 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.  =  <. z ,  ( Q  |`  pred ( z ,  A ,  R
) ) >. )
5958, 28syl6eqr 2485 . . . . . . . . . . 11  |-  ( f  =  Q  ->  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.  =  W )
6059fveq2d 5724 . . . . . . . . . 10  |-  ( f  =  Q  ->  ( G `  <. z ,  ( f  |`  pred (
z ,  A ,  R ) ) >.
)  =  ( G `
 W ) )
6156, 60eqeq12d 2449 . . . . . . . . 9  |-  ( f  =  Q  ->  (
( f `  z
)  =  ( G `
 <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )  <->  ( Q `  z )  =  ( G `  W ) ) )
6261ralbidv 2717 . . . . . . . 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 692 . . . . . . 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 2716 . . . . . 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 29154 . . . . 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 224 . . 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 5720 . . . . . . . 8  |-  ( x  =  z  ->  (
f `  x )  =  ( f `  z ) )
69 id 20 . . . . . . . . . . 11  |-  ( x  =  z  ->  x  =  z )
70 bnj602 29223 . . . . . . . . . . . 12  |-  ( x  =  z  ->  pred (
x ,  A ,  R )  =  pred ( z ,  A ,  R ) )
7170reseq2d 5138 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
f  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( z ,  A ,  R ) ) )
7269, 71opeq12d 3984 . . . . . . . . . 10  |-  ( x  =  z  ->  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
7313, 72syl5eq 2479 . . . . . . . . 9  |-  ( x  =  z  ->  Y  =  <. z ,  ( f  |`  pred ( z ,  A ,  R
) ) >. )
7473fveq2d 5724 . . . . . . . 8  |-  ( x  =  z  ->  ( G `  Y )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >. ) )
7568, 74eqeq12d 2449 . . . . . . 7  |-  ( x  =  z  ->  (
( f `  x
)  =  ( G `
 Y )  <->  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) ) )
7675cbvralv 2924 . . . . . 6  |-  ( A. x  e.  d  (
f `  x )  =  ( G `  Y )  <->  A. z  e.  d  ( f `  z )  =  ( G `  <. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
) )
7776anbi2i 676 . . . . 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 2722 . . . 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 3208 . . 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 204 . 2  |-  ( ch 
->  [. Q  /  f ]. E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
8114bnj1454 29150 . . 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 224 1  |-  ( ch 
->  Q  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948   [.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    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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-bnj14 28990
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