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Theorem bnj1307 29098
Description: Technical lemma for bnj60 29137. 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
bnj1307.1  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1307.2  |-  ( w  e.  B  ->  A. x  w  e.  B )
Assertion
Ref Expression
bnj1307  |-  ( w  e.  C  ->  A. x  w  e.  C )
Distinct variable groups:    w, B    w, d, x    x, f
Allowed substitution hints:    B( x, f, d)    C( x, w, f, d)    G( x, w, f, d)    Y( x, w, f, d)

Proof of Theorem bnj1307
StepHypRef Expression
1 bnj1307.1 . . 3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
2 bnj1307.2 . . . . . 6  |-  ( w  e.  B  ->  A. x  w  e.  B )
32nfcii 2531 . . . . 5  |-  F/_ x B
4 nfv 1626 . . . . . 6  |-  F/ x  f  Fn  d
5 nfra1 2716 . . . . . 6  |-  F/ x A. x  e.  d 
( f `  x
)  =  ( G `
 Y )
64, 5nfan 1842 . . . . 5  |-  F/ x
( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
73, 6nfrex 2721 . . . 4  |-  F/ x E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
87nfab 2544 . . 3  |-  F/_ x { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
91, 8nfcxfr 2537 . 2  |-  F/_ x C
109nfcrii 2533 1  |-  ( w  e.  C  ->  A. x  w  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  bnj1311  29099  bnj1373  29105  bnj1498  29136  bnj1525  29144  bnj1523  29146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672
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