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Theorem bnj911 29241
Description: Technical lemma for bnj69 29317. 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
bnj911.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj911.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj911  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  A. i
( f  Fn  n  /\  ph  /\  ps )
)
Distinct variable groups:    f, i    i, n    ph, i
Allowed substitution hints:    ph( y, f, n)    ps( y, f, i, n)    A( y, f, i, n)    R( y, f, i, n)    X( y, f, i, n)

Proof of Theorem bnj911
StepHypRef Expression
1 bnj911.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
21bnj1095 29090 . 2  |-  ( ps 
->  A. i ps )
32bnj1350 29135 1  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  A. i
( f  Fn  n  /\  ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2698   (/)c0 3621   U_ciun 4086   suc csuc 4576   omcom 4838    Fn wfn 5442   ` cfv 5447    predc-bnj14 28990
This theorem is referenced by:  bnj916  29242  bnj1014  29269
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-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1551  df-nf 1554  df-ral 2703
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