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Theorem bnj1514 29433
Description: Technical lemma for bnj1500 29438. 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
bnj1514.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1514.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1514.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
Assertion
Ref Expression
bnj1514  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
Distinct variable groups:    x, A    G, d    Y, d    f, d, x
Allowed substitution hints:    A( f, d)    B( x, f, d)    C( x, f, d)    R( x, f, d)    G( x, f)    Y( x, f)

Proof of Theorem bnj1514
StepHypRef Expression
1 bnj1514.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
21bnj1436 29212 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
3 df-rex 2759 . . . . 5  |-  ( E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  E. d
( d  e.  B  /\  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) ) )
4 3anass 978 . . . . 5  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  ( d  e.  B  /\  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
53, 4bnj133 29094 . . . 4  |-  ( E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  <->  E. d
( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
62, 5sylib 196 . . 3  |-  ( f  e.  C  ->  E. d
( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
7 simp3 999 . . . 4  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  ->  A. x  e.  d 
( f `  x
)  =  ( G `
 Y ) )
8 fndm 5660 . . . . . 6  |-  ( f  Fn  d  ->  dom  f  =  d )
983ad2ant2 1019 . . . . 5  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  ->  dom  f  =  d
)
109raleqdv 3009 . . . 4  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  -> 
( A. x  e. 
dom  f ( f `
 x )  =  ( G `  Y
)  <->  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
117, 10mpbird 232 . . 3  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  ->  A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
126, 11bnj593 29116 . 2  |-  ( f  e.  C  ->  E. d A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
1312bnj937 29144 1  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   A.wral 2753   E.wrex 2754    C_ wss 3413   <.cop 3977   dom cdm 4822    |` cres 4824    Fn wfn 5563   ` cfv 5568    predc-bnj14 29054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-fn 5571
This theorem is referenced by:  bnj1501  29437
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