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Theorem bnj1514 33198
Description: Technical lemma for bnj1500 33203. 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 32977 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
3 df-rex 2820 . . . . 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 977 . . . . 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 32860 . . . 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 998 . . . 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 5678 . . . . . 6  |-  ( f  Fn  d  ->  dom  f  =  d )
983ad2ant2 1018 . . . . 5  |-  ( ( d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) )  ->  dom  f  =  d
)
109raleqdv 3064 . . . 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 32881 . 2  |-  ( f  e.  C  ->  E. d A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
1312bnj937 32909 1  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815    C_ wss 3476   <.cop 4033   dom cdm 4999    |` cres 5001    Fn wfn 5581   ` cfv 5586    predc-bnj14 32820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-fn 5589
This theorem is referenced by:  bnj1501  33202
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