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Theorem bnj579 29262
Description: Technical lemma for bnj852 29269. 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 29261 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 1632    e. wcel 1696   E*wmo 2157   A.wral 2556   [.wsbc 3004    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   class class class wbr 4039    _E cep 4319   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj600  29267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-bnj17 29028
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