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Theorem bnj580 31835
Description: Technical lemma for bnj579 31836. 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 1003 . . . . . 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 31830 . . . . . . 7  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
76simp1bi 1003 . . . . . 6  |-  ( ch'  ->  g  Fn  n )
82, 7bnj240 31616 . . . . 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 31800 . . . . . . . . . . . . 13  |-  ( ph'  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
13 vex 2970 . . . . . . . . . . . . . 14  |-  g  e. 
_V
1410, 4, 13bnj540 31814 . . . . . . . . . . . . 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 31833 . . . . . . . . . . . . 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 31834 . . . . . . . . . . . 12  |-  ( ( j  e.  n  /\  ta )  ->  th )
1918ex 434 . . . . . . . . . . 11  |-  ( j  e.  n  ->  ( ta  ->  th ) )
2019rgen 2776 . . . . . . . . . 10  |-  A. j  e.  n  ( ta  ->  th )
21 vex 2970 . . . . . . . . . . 11  |-  n  e. 
_V
2221, 17bnj110 31780 . . . . . . . . . 10  |-  ( (  _E  Fr  n  /\  A. j  e.  n  ( ta  ->  th )
)  ->  A. j  e.  n  th )
2320, 22mpan2 671 . . . . . . . . 9  |-  (  _E  Fr  n  ->  A. j  e.  n  th )
2415ralbii 2734 . . . . . . . . 9  |-  ( A. j  e.  n  th  <->  A. j  e.  n  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
2523, 24sylib 196 . . . . . . . 8  |-  (  _E  Fr  n  ->  A. j  e.  n  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
2625r19.21be 2812 . . . . . . 7  |-  A. j  e.  n  (  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
2711bnj923 31690 . . . . . . . . . . . . 13  |-  ( n  e.  D  ->  n  e.  om )
28 nnord 6479 . . . . . . . . . . . . 13  |-  ( n  e.  om  ->  Ord  n )
29 ordfr 4729 . . . . . . . . . . . . 13  |-  ( Ord  n  ->  _E  Fr  n )
3027, 28, 293syl 20 . . . . . . . . . . . 12  |-  ( n  e.  D  ->  _E  Fr  n )
31303ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  _E  Fr  n )
3231pm4.71ri 633 . . . . . . . . . 10  |-  ( ( n  e.  D  /\  ch  /\  ch' )  <->  (  _E  Fr  n  /\  (
n  e.  D  /\  ch  /\  ch' ) ) )
3332imbi1i 325 . . . . . . . . 9  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
( (  _E  Fr  n  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  (
f `  j )  =  ( g `  j ) ) )
34 impexp 446 . . . . . . . . 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 249 . . . . . . . 8  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
(  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) ) )
3635ralbii 2734 . . . . . . 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 209 . . . . . 6  |-  A. j  e.  n  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) )
38 r19.21v 2798 . . . . . 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 208 . . . . 5  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  A. j  e.  n  ( f `  j )  =  ( g `  j ) )
40 eqfnfv 5792 . . . . . 6  |-  ( ( f  Fn  n  /\  g  Fn  n )  ->  ( f  =  g  <->  A. j  e.  n  ( f `  j
)  =  ( g `
 j ) ) )
4140biimprd 223 . . . . 5  |-  ( ( f  Fn  n  /\  g  Fn  n )  ->  ( A. j  e.  n  ( f `  j )  =  ( g `  j )  ->  f  =  g ) )
428, 39, 41sylc 60 . . . 4  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  f  =  g )
43423expib 1190 . . 3  |-  ( n  e.  D  ->  (
( ch  /\  ch' )  -> 
f  =  g ) )
4443alrimivv 1686 . 2  |-  ( n  e.  D  ->  A. f A. g ( ( ch 
/\  ch' )  ->  f  =  g ) )
45 sbsbc 3185 . . . . . 6  |-  ( [ g  /  f ] ch  <->  [. g  /  f ]. ch )
4645anbi2i 694 . . . . 5  |-  ( ( ch  /\  [ g  /  f ] ch ) 
<->  ( ch  /\  [. g  /  f ]. ch ) )
4746imbi1i 325 . . . 4  |-  ( ( ( ch  /\  [
g  /  f ] ch )  ->  f  =  g )  <->  ( ( ch  /\  [. g  / 
f ]. ch )  -> 
f  =  g ) )
48472albii 1611 . . 3  |-  ( A. f A. g ( ( ch  /\  [ g  /  f ] ch )  ->  f  =  g )  <->  A. f A. g
( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
49 nfv 1673 . . . 4  |-  F/ g ch
5049mo3 2296 . . 3  |-  ( E* f ch  <->  A. f A. g ( ( ch 
/\  [ g  / 
f ] ch )  ->  f  =  g ) )
515anbi2i 694 . . . . 5  |-  ( ( ch  /\  ch' )  <->  ( ch  /\ 
[. g  /  f ]. ch ) )
5251imbi1i 325 . . . 4  |-  ( ( ( ch  /\  ch' )  -> 
f  =  g )  <-> 
( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
53522albii 1611 . . 3  |-  ( A. f A. g ( ( ch  /\  ch' )  -> 
f  =  g )  <->  A. f A. g ( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
5448, 50, 533bitr4i 277 . 2  |-  ( E* f ch  <->  A. f A. g ( ( ch 
/\  ch' )  ->  f  =  g ) )
5544, 54sylibr 212 1  |-  ( n  e.  D  ->  E* f ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   [wsb 1700    e. wcel 1756   E*wmo 2253   A.wral 2710   [.wsbc 3181    \ cdif 3320   (/)c0 3632   {csn 3872   U_ciun 4166   class class class wbr 4287    _E cep 4625    Fr wfr 4671   Ord word 4713   suc csuc 4716    Fn wfn 5408   ` cfv 5413   omcom 6471    predc-bnj14 31605
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-om 6472  df-bnj17 31604
This theorem is referenced by:  bnj579  31836
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