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Theorem bnj579 29285
Description: Technical lemma for bnj852 29292. 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 228 . 2  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) )
4 biid 228 . 2  |-  ( [. g  /  f ]. ph  <->  [. g  / 
f ]. ph )
5 biid 228 . 2  |-  ( [. g  /  f ]. ps  <->  [. g  /  f ]. ps )
6 biid 228 . 2  |-  ( [. g  /  f ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  [. g  / 
f ]. ( f  Fn  n  /\  ph  /\  ps ) )
7 bnj579.3 . 2  |-  D  =  ( om  \  { (/)
} )
8 biid 228 . 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 228 . 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 29284 1  |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   E*wmo 2282   A.wral 2705   [.wsbc 3161    \ cdif 3317   (/)c0 3628   {csn 3814   U_ciun 4093   class class class wbr 4212    _E cep 4492   suc csuc 4583   omcom 4845    Fn wfn 5449   ` cfv 5454    predc-bnj14 29052
This theorem is referenced by:  bnj600  29290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-bnj17 29051
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