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Theorem bnj580 28990
Description: Technical lemma for bnj579 28991. 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 972 . . . . . 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 28985 . . . . . . 7  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
76simp1bi 972 . . . . . 6  |-  ( ch'  ->  g  Fn  n )
82, 7bnj240 28769 . . . . 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 28955 . . . . . . . . . . . . 13  |-  ( ph'  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
13 vex 2919 . . . . . . . . . . . . . 14  |-  g  e. 
_V
1410, 4, 13bnj540 28969 . . . . . . . . . . . . 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 28988 . . . . . . . . . . . . 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 28989 . . . . . . . . . . . 12  |-  ( ( j  e.  n  /\  ta )  ->  th )
1918ex 424 . . . . . . . . . . 11  |-  ( j  e.  n  ->  ( ta  ->  th ) )
2019rgen 2731 . . . . . . . . . 10  |-  A. j  e.  n  ( ta  ->  th )
21 vex 2919 . . . . . . . . . . 11  |-  n  e. 
_V
2221, 17bnj110 28935 . . . . . . . . . 10  |-  ( (  _E  Fr  n  /\  A. j  e.  n  ( ta  ->  th )
)  ->  A. j  e.  n  th )
2320, 22mpan2 653 . . . . . . . . 9  |-  (  _E  Fr  n  ->  A. j  e.  n  th )
2415ralbii 2690 . . . . . . . . 9  |-  ( A. j  e.  n  th  <->  A. j  e.  n  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
2523, 24sylib 189 . . . . . . . 8  |-  (  _E  Fr  n  ->  A. j  e.  n  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
2625r19.21be 2767 . . . . . . 7  |-  A. j  e.  n  (  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
2711bnj923 28843 . . . . . . . . . . . . 13  |-  ( n  e.  D  ->  n  e.  om )
28 nnord 4812 . . . . . . . . . . . . 13  |-  ( n  e.  om  ->  Ord  n )
29 ordfr 4556 . . . . . . . . . . . . 13  |-  ( Ord  n  ->  _E  Fr  n )
3027, 28, 293syl 19 . . . . . . . . . . . 12  |-  ( n  e.  D  ->  _E  Fr  n )
31303ad2ant1 978 . . . . . . . . . . 11  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  _E  Fr  n )
3231pm4.71ri 615 . . . . . . . . . 10  |-  ( ( n  e.  D  /\  ch  /\  ch' )  <->  (  _E  Fr  n  /\  (
n  e.  D  /\  ch  /\  ch' ) ) )
3332imbi1i 316 . . . . . . . . 9  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
( (  _E  Fr  n  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  (
f `  j )  =  ( g `  j ) ) )
34 impexp 434 . . . . . . . . 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 241 . . . . . . . 8  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
(  _E  Fr  n  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) ) )
3635ralbii 2690 . . . . . . 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 201 . . . . . 6  |-  A. j  e.  n  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) )
38 r19.21v 2753 . . . . . 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 200 . . . . 5  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  A. j  e.  n  ( f `  j )  =  ( g `  j ) )
40 eqfnfv 5786 . . . . . 6  |-  ( ( f  Fn  n  /\  g  Fn  n )  ->  ( f  =  g  <->  A. j  e.  n  ( f `  j
)  =  ( g `
 j ) ) )
4140biimprd 215 . . . . 5  |-  ( ( f  Fn  n  /\  g  Fn  n )  ->  ( A. j  e.  n  ( f `  j )  =  ( g `  j )  ->  f  =  g ) )
428, 39, 41sylc 58 . . . 4  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  f  =  g )
43423expib 1156 . . 3  |-  ( n  e.  D  ->  (
( ch  /\  ch' )  -> 
f  =  g ) )
4443alrimivv 1639 . 2  |-  ( n  e.  D  ->  A. f A. g ( ( ch 
/\  ch' )  ->  f  =  g ) )
45 sbsbc 3125 . . . . . 6  |-  ( [ g  /  f ] ch  <->  [. g  /  f ]. ch )
4645anbi2i 676 . . . . 5  |-  ( ( ch  /\  [ g  /  f ] ch ) 
<->  ( ch  /\  [. g  /  f ]. ch ) )
4746imbi1i 316 . . . 4  |-  ( ( ( ch  /\  [
g  /  f ] ch )  ->  f  =  g )  <->  ( ( ch  /\  [. g  / 
f ]. ch )  -> 
f  =  g ) )
48472albii 1573 . . 3  |-  ( A. f A. g ( ( ch  /\  [ g  /  f ] ch )  ->  f  =  g )  <->  A. f A. g
( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
49 nfv 1626 . . . 4  |-  F/ g ch
5049mo3 2285 . . 3  |-  ( E* f ch  <->  A. f A. g ( ( ch 
/\  [ g  / 
f ] ch )  ->  f  =  g ) )
515anbi2i 676 . . . . 5  |-  ( ( ch  /\  ch' )  <->  ( ch  /\ 
[. g  /  f ]. ch ) )
5251imbi1i 316 . . . 4  |-  ( ( ( ch  /\  ch' )  -> 
f  =  g )  <-> 
( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
53522albii 1573 . . 3  |-  ( A. f A. g ( ( ch  /\  ch' )  -> 
f  =  g )  <->  A. f A. g ( ( ch  /\  [. g  /  f ]. ch )  ->  f  =  g ) )
5448, 50, 533bitr4i 269 . 2  |-  ( E* f ch  <->  A. f A. g ( ( ch 
/\  ch' )  ->  f  =  g ) )
5544, 54sylibr 204 1  |-  ( n  e.  D  ->  E* f ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649   [wsb 1655    e. wcel 1721   E*wmo 2255   A.wral 2666   [.wsbc 3121    \ cdif 3277   (/)c0 3588   {csn 3774   U_ciun 4053   class class class wbr 4172    _E cep 4452    Fr wfr 4498   Ord word 4540   suc csuc 4543   omcom 4804    Fn wfn 5408   ` cfv 5413    predc-bnj14 28758
This theorem is referenced by:  bnj579  28991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-bnj17 28757
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