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Theorem bnj1445 32317
Description: Technical lemma for bnj60 32335. 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
bnj1445.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1445.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1445.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1445.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1445.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1445.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1445.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1445.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1445.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1445.10  |-  P  = 
U. H
bnj1445.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1445.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1445.13  |-  W  = 
<. z ,  ( Q  |`  pred ( z ,  A ,  R ) ) >.
bnj1445.14  |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
bnj1445.15  |-  ( ch 
->  P  Fn  trCl (
x ,  A ,  R ) )
bnj1445.16  |-  ( ch 
->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1445.17  |-  ( th  <->  ( ch  /\  z  e.  E ) )
bnj1445.18  |-  ( et  <->  ( th  /\  z  e. 
{ x } ) )
bnj1445.19  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
bnj1445.20  |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e. 
dom  f ) )
bnj1445.21  |-  ( si  <->  ( rh  /\  y  e. 
pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
bnj1445.22  |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
bnj1445.23  |-  X  = 
<. z ,  ( f  |`  pred ( z ,  A ,  R ) ) >.
Assertion
Ref Expression
bnj1445  |-  ( si  ->  A. d si )
Distinct variable groups:    A, d, x    B, f    E, d    R, d, x    f, d, x    y, d, x   
z, d
Allowed substitution hints:    ph( x, y, z, f, d)    ps( x, y, z, f, d)    ch( x, y, z, f, d)    th( x, y, z, f, d)    ta( x, y, z, f, d)    et( x, y, z, f, d)    ze( x, y, z, f, d)    si( x, y, z, f, d)    rh( x, y, z, f, d)    A( y, z, f)    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, z, f, d)    R( y, z, f)    E( x, y, z, f)    G( x, y, z, f, d)    H( x, y, z, f, d)    W( x, y, z, f, d)    X( x, y, z, f, d)    Y( x, y, z, f, d)    Z( x, y, z, f, d)    ta'( x, y, z, f, d)

Proof of Theorem bnj1445
StepHypRef Expression
1 bnj1445.21 . 2  |-  ( si  <->  ( rh  /\  y  e. 
pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
2 bnj1445.20 . . . . 5  |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e. 
dom  f ) )
3 bnj1445.19 . . . . . . 7  |-  ( ze  <->  ( th  /\  z  e. 
trCl ( x ,  A ,  R ) ) )
4 bnj1445.17 . . . . . . . . 9  |-  ( th  <->  ( ch  /\  z  e.  E ) )
5 bnj1445.7 . . . . . . . . . . . . 13  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
6 bnj1445.6 . . . . . . . . . . . . . . 15  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
7 nfv 1674 . . . . . . . . . . . . . . . 16  |-  F/ d  R  FrSe  A
8 bnj1445.5 . . . . . . . . . . . . . . . . . 18  |-  D  =  { x  e.  A  |  -.  E. f ta }
9 bnj1445.4 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
10 bnj1445.3 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
11 nfre1 2858 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
1211nfab 2614 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
1310, 12nfcxfr 2608 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  F/_ d C
1413nfcri 2603 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ d  f  e.  C
15 nfv 1674 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ d dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
1614, 15nfan 1862 . . . . . . . . . . . . . . . . . . . . . 22  |-  F/ d ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
179, 16nfxfr 1616 . . . . . . . . . . . . . . . . . . . . 21  |-  F/ d ta
1817nfex 1875 . . . . . . . . . . . . . . . . . . . 20  |-  F/ d E. f ta
1918nfn 1836 . . . . . . . . . . . . . . . . . . 19  |-  F/ d  -.  E. f ta
20 nfcv 2610 . . . . . . . . . . . . . . . . . . 19  |-  F/_ d A
2119, 20nfrab 2984 . . . . . . . . . . . . . . . . . 18  |-  F/_ d { x  e.  A  |  -.  E. f ta }
228, 21nfcxfr 2608 . . . . . . . . . . . . . . . . 17  |-  F/_ d D
23 nfcv 2610 . . . . . . . . . . . . . . . . 17  |-  F/_ d (/)
2422, 23nfne 2776 . . . . . . . . . . . . . . . 16  |-  F/ d  D  =/=  (/)
257, 24nfan 1862 . . . . . . . . . . . . . . 15  |-  F/ d ( R  FrSe  A  /\  D  =/=  (/) )
266, 25nfxfr 1616 . . . . . . . . . . . . . 14  |-  F/ d ps
2722nfcri 2603 . . . . . . . . . . . . . 14  |-  F/ d  x  e.  D
28 nfv 1674 . . . . . . . . . . . . . . 15  |-  F/ d  -.  y R x
2922, 28nfral 2855 . . . . . . . . . . . . . 14  |-  F/ d A. y  e.  D  -.  y R x
3026, 27, 29nf3an 1864 . . . . . . . . . . . . 13  |-  F/ d ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x )
315, 30nfxfr 1616 . . . . . . . . . . . 12  |-  F/ d ch
3231nfri 1809 . . . . . . . . . . 11  |-  ( ch 
->  A. d ch )
3332bnj1351 32102 . . . . . . . . . 10  |-  ( ( ch  /\  z  e.  E )  ->  A. d
( ch  /\  z  e.  E ) )
3433nfi 1597 . . . . . . . . 9  |-  F/ d ( ch  /\  z  e.  E )
354, 34nfxfr 1616 . . . . . . . 8  |-  F/ d th
36 nfv 1674 . . . . . . . 8  |-  F/ d  z  e.  trCl (
x ,  A ,  R )
3735, 36nfan 1862 . . . . . . 7  |-  F/ d ( th  /\  z  e.  trCl ( x ,  A ,  R ) )
383, 37nfxfr 1616 . . . . . 6  |-  F/ d ze
39 bnj1445.9 . . . . . . . 8  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
40 nfcv 2610 . . . . . . . . . 10  |-  F/_ d  pred ( x ,  A ,  R )
41 bnj1445.8 . . . . . . . . . . 11  |-  ( ta'  <->  [. y  /  x ]. ta )
42 nfcv 2610 . . . . . . . . . . . 12  |-  F/_ d
y
4342, 17nfsbc 3292 . . . . . . . . . . 11  |-  F/ d
[. y  /  x ]. ta
4441, 43nfxfr 1616 . . . . . . . . . 10  |-  F/ d ta'
4540, 44nfrex 2857 . . . . . . . . 9  |-  F/ d E. y  e.  pred  ( x ,  A ,  R ) ta'
4645nfab 2614 . . . . . . . 8  |-  F/_ d { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
4739, 46nfcxfr 2608 . . . . . . 7  |-  F/_ d H
4847nfcri 2603 . . . . . 6  |-  F/ d  f  e.  H
49 nfv 1674 . . . . . 6  |-  F/ d  z  e.  dom  f
5038, 48, 49nf3an 1864 . . . . 5  |-  F/ d ( ze  /\  f  e.  H  /\  z  e.  dom  f )
512, 50nfxfr 1616 . . . 4  |-  F/ d rh
5251nfri 1809 . . 3  |-  ( rh 
->  A. d rh )
53 ax-5 1671 . . 3  |-  ( y  e.  pred ( x ,  A ,  R )  ->  A. d  y  e. 
pred ( x ,  A ,  R ) )
5414nfri 1809 . . 3  |-  ( f  e.  C  ->  A. d 
f  e.  C )
55 ax-5 1671 . . 3  |-  ( dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) )  ->  A. d dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
5652, 53, 54, 55bnj982 32054 . 2  |-  ( ( rh  /\  y  e. 
pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )  ->  A. d ( rh 
/\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) ) )
571, 56hbxfrbi 1614 1  |-  ( si  ->  A. d si )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1757   {cab 2435    =/= wne 2641   A.wral 2792   E.wrex 2793   {crab 2796   [.wsbc 3270    u. cun 3410    C_ wss 3412   (/)c0 3721   {csn 3961   <.cop 3967   U.cuni 4175   class class class wbr 4376   dom cdm 4924    |` cres 4926    Fn wfn 5497   ` cfv 5502    /\ w-bnj17 31956    predc-bnj14 31958    FrSe w-bnj15 31962    trClc-bnj18 31964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-sbc 3271  df-bnj17 31957
This theorem is referenced by:  bnj1450  32323
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