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Theorem bnj580 29736
Description: Technical lemma for bnj579 29737. 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
bnj580.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj580.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj580.3  |-  ( ch  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj580.4  |-  ( ph'  <->  [. g  /  f ]. ph )
bnj580.5  |-  ( ps'  <->  [. g  /  f ]. ps )
bnj580.6  |-  ( ch'  <->  [. g  /  f ]. ch )
bnj580.7  |-  D  =  ( om  \  { (/)
} )
bnj580.8  |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
bnj580.9  |-  ( ta  <->  A. k  e.  n  ( k  _E  j  ->  [. k  /  j ]. th ) )
Assertion
Ref Expression
bnj580  |-  ( n  e.  D  ->  E* f ch )
Distinct variable groups:    A, f,
i, k    D, f,
g, j, k    R, f, i, k    ch, g,
j, k    j, ch', k    f, n    g, i, n, k   
x, f    y, f,
g, i, k    j, n    th, k
Allowed substitution hints:    ph( x, y, f, g, i, j, k, n)    ps( x, y, f, g, i, j, k, n)    ch( x, y, f, i, n)    th( x, y, f, g, i, j, n)    ta( x, y, f, g, i, j, k, n)    A( x, y, g, j, n)    D( x, y, i, n)    R( x, y, g, j, n)    ph'( x, y, f, g, i, j, k, n)    ps'( x, y, f, g, i, j, k, n)    ch'( x, y, f, g, i, n)

Proof of Theorem bnj580
StepHypRef Expression
1 bnj580.3 . . . . . . 7  |-  ( ch  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
21simp1bi 1024 . . . . . 6  |-  ( ch 
->  f  Fn  n
)
3 bnj580.4 . . . . . . . 8  |-  ( ph'  <->  [. g  /  f ]. ph )
4 bnj580.5 . . . . . . . 8  |-  ( ps'  <->  [. g  /  f ]. ps )
5 bnj580.6 . . . . . . . 8  |-  ( ch'  <->  [. g  /  f ]. ch )
61, 3, 4, 5bnj581 29731 . . . . . . 7  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
76simp1bi 1024 . . . . . 6  |-  ( ch'  ->  g  Fn  n )
82, 7bnj240 29516 . . . . 5  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f  Fn  n  /\  g  Fn  n )
)
9 bnj580.1 . . . . . . . . . . . . 13  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
10 bnj580.2 . . . . . . . . . . . . 13  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
11 bnj580.7 . . . . . . . . . . . . 13  |-  D  =  ( om  \  { (/)
} )
123, 9bnj154 29701 . . . . . . . . . . . . 13  |-  ( ph'  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
13 vex 3050 . . . . . . . . . . . . . 14  |-  g  e. 
_V
1410, 4, 13bnj540 29715 . . . . . . . . . . . . 13  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) )
15 bnj580.8 . . . . . . . . . . . . 13  |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
1615bnj591 29734 . . . . . . . . . . . . 13  |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  k
)  =  ( g `
 k ) ) )
17 bnj580.9 . . . . . . . . . . . . 13  |-  ( ta  <->  A. k  e.  n  ( k  _E  j  ->  [. k  /  j ]. th ) )
189, 10, 1, 11, 12, 14, 6, 15, 16, 17bnj594 29735 . . . . . . . . . . . 12  |-  ( ( j  e.  n  /\  ta )  ->  th )
1918ex 436 . . . . . . . . . . 11  |-  ( j  e.  n  ->  ( ta  ->  th ) )
2019rgen 2749 . . . . . . . . . 10  |-  A. j  e.  n  ( ta  ->  th )
21 vex 3050 . . . . . . . . . . 11  |-  n  e. 
_V
2221, 17bnj110 29681 . . . . . . . . . 10  |-  ( (  _E  Fr  n  /\  A. j  e.  n  ( ta  ->  th )
)  ->  A. j  e.  n  th )
2320, 22mpan2 678 . . . . . . . . 9  |-  (  _E  Fr  n  ->  A. j  e.  n  th )
2415ralbii 2821 . . . . . . . . 9  |-  ( A. j  e.  n  th  <->  A. j  e.  n  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
2523, 24sylib 200 . . . . . . . 8  |-  (  _E  Fr  n  ->  A. j  e.  n  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
2625r19.21be 2761 . . . . . . 7  |-  A. j  e.  n  (  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
2711bnj923 29591 . . . . . . . . . . . . 13  |-  ( n  e.  D  ->  n  e.  om )
28 nnord 6705 . . . . . . . . . . . . 13  |-  ( n  e.  om  ->  Ord  n )
29 ordfr 5441 . . . . . . . . . . . . 13  |-  ( Ord  n  ->  _E  Fr  n )
3027, 28, 293syl 18 . . . . . . . . . . . 12  |-  ( n  e.  D  ->  _E  Fr  n )
31303ad2ant1 1030 . . . . . . . . . . 11  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  _E  Fr  n )
3231pm4.71ri 639 . . . . . . . . . 10  |-  ( ( n  e.  D  /\  ch  /\  ch' )  <->  (  _E  Fr  n  /\  (
n  e.  D  /\  ch  /\  ch' ) ) )
3332imbi1i 327 . . . . . . . . 9  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
( (  _E  Fr  n  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  (
f `  j )  =  ( g `  j ) ) )
34 impexp 448 . . . . . . . . 9  |-  ( ( (  _E  Fr  n  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( f `  j )  =  ( g `  j ) )  <->  (  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `
 j )  =  ( g `  j
) ) ) )
3533, 34bitri 253 . . . . . . . 8  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
(  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) ) )
3635ralbii 2821 . . . . . . 7  |-  ( A. j  e.  n  (
( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <->  A. j  e.  n  (  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) ) )
3726, 36mpbir 213 . . . . . 6  |-  A. j  e.  n  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) )
38 r19.21v 2795 . . . . . 6  |-  ( A. j  e.  n  (
( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
( ( n  e.  D  /\  ch  /\  ch' )  ->  A. j  e.  n  ( f `  j )  =  ( g `  j ) ) )
3937, 38mpbi 212 . . . . 5  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  A. j  e.  n  ( f `  j )  =  ( g `  j ) )
40 eqfnfv 5981 . . . . . 6  |-  ( ( f  Fn  n  /\  g  Fn  n )  ->  ( f  =  g  <->  A. j  e.  n  ( f `  j
)  =  ( g `
 j ) ) )
4140biimprd 227 . . . . 5  |-  ( ( f  Fn  n  /\  g  Fn  n )  ->  ( A. j  e.  n  ( f `  j )  =  ( g `  j )  ->  f  =  g ) )
428, 39, 41sylc 62 . . . 4  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  f  =  g )
43423expib 1212 . . 3  |-  ( n  e.  D  ->  (
( ch  /\  ch' )  -> 
f  =  g ) )
4443alrimivv 1776 . 2  |-  ( n  e.  D  ->  A. f A. g ( ( ch 
/\  ch' )  ->  f  =  g ) )
45 sbsbc 3273 . . . . . 6  |-  ( [ g  /  f ] ch  <->  [. g  /  f ]. ch )
4645anbi2i 701 . . . . 5  |-  ( ( ch  /\  [ g  /  f ] ch ) 
<->  ( ch  /\  [. g  /  f ]. ch ) )
4746imbi1i 327 . . . 4  |-  ( ( ( ch  /\  [
g  /  f ] ch )  ->  f  =  g )  <->  ( ( ch  /\  [. g  / 
f ]. ch )  -> 
f  =  g ) )
48472albii 1694 . . 3  |-  ( A. f A. g ( ( ch  /\  [ g  /  f ] ch )  ->  f  =  g )  <->  A. f A. g
( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
49 nfv 1763 . . . 4  |-  F/ g ch
5049mo3 2338 . . 3  |-  ( E* f ch  <->  A. f A. g ( ( ch 
/\  [ g  / 
f ] ch )  ->  f  =  g ) )
515anbi2i 701 . . . . 5  |-  ( ( ch  /\  ch' )  <->  ( ch  /\ 
[. g  /  f ]. ch ) )
5251imbi1i 327 . . . 4  |-  ( ( ( ch  /\  ch' )  -> 
f  =  g )  <-> 
( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
53522albii 1694 . . 3  |-  ( A. f A. g ( ( ch  /\  ch' )  -> 
f  =  g )  <->  A. f A. g ( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
5448, 50, 533bitr4i 281 . 2  |-  ( E* f ch  <->  A. f A. g ( ( ch 
/\  ch' )  ->  f  =  g ) )
5544, 54sylibr 216 1  |-  ( n  e.  D  ->  E* f ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986   A.wal 1444    = wceq 1446   [wsb 1799    e. wcel 1889   E*wmo 2302   A.wral 2739   [.wsbc 3269    \ cdif 3403   (/)c0 3733   {csn 3970   U_ciun 4281   class class class wbr 4405    _E cep 4746    Fr wfr 4793   Ord word 5425   suc csuc 5428    Fn wfn 5580   ` cfv 5585   omcom 6697    predc-bnj14 29505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-om 6698  df-bnj17 29504
This theorem is referenced by:  bnj579  29737
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