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Theorem bnj579 28946
Description: Technical lemma for bnj852 28953. 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
bnj579.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj579.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj579.3  |-  D  =  ( om  \  { (/)
} )
Assertion
Ref Expression
bnj579  |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
Distinct variable groups:    A, f,
i    D, f    R, f, i    f, n, i   
x, f    y, f,
i
Allowed substitution hints:    ph( x, y, f, i, n)    ps( x, y, f, i, n)    A( x, y, n)    D( x, y, i, n)    R( x, y, n)

Proof of Theorem bnj579
Dummy variables  k 
g  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj579.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
2 bnj579.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 biid 227 . 2  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
4 biid 227 . 2  |-  ( [. g  /  f ]. ph  <->  [. g  / 
f ]. ph )
5 biid 227 . 2  |-  ( [. g  /  f ]. ps  <->  [. g  /  f ]. ps )
6 biid 227 . 2  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  [. g  / 
f ]. ( f  Fn  n  /\  ph  /\  ps ) )
7 bnj579.3 . 2  |-  D  =  ( om  \  { (/)
} )
8 biid 227 . 2  |-  ( ( ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )  /\  [. g  /  f ]. ( f  Fn  n  /\  ph  /\  ps )
)  ->  ( f `  j )  =  ( g `  j ) )  <->  ( ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps )  /\  [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps ) )  -> 
( f `  j
)  =  ( g `
 j ) ) )
9 biid 227 . 2  |-  ( A. k  e.  n  (
k  _E  j  ->  [. k  /  j ]. ( ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\ 
ps )  /\  [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps ) )  -> 
( f `  j
)  =  ( g `
 j ) ) )  <->  A. k  e.  n  ( k  _E  j  ->  [. k  /  j ]. ( ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\ 
ps )  /\  [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps ) )  -> 
( f `  j
)  =  ( g `
 j ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9bnj580 28945 1  |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   E*wmo 2144   A.wral 2543   [.wsbc 2991    \ cdif 3149   (/)c0 3455   {csn 3640   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj600  28951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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