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Theorem bnj1514 29432
Description: Technical lemma for bnj1500 29437. 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 29211 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
3 df-rex 2711 . . . . 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 940 . . . . 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 29092 . . . 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 189 . . 3  |-  ( f  e.  C  ->  E. d
( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) )
7 simp3 959 . . . 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 5544 . . . . . 6  |-  ( f  Fn  d  ->  dom  f  =  d )
983ad2ant2 979 . . . . 5  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  ->  dom  f  =  d
)
109raleqdv 2910 . . . 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 224 . . 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 29113 . 2  |-  ( f  e.  C  ->  E. d A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
1312bnj937 29142 1  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706    C_ wss 3320   <.cop 3817   dom cdm 4878    |` cres 4880    Fn wfn 5449   ` cfv 5454    predc-bnj14 29052
This theorem is referenced by:  bnj1501  29436
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-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-fn 5457
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