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Theorem bnj1307 29490
Description: Technical lemma for bnj60 29529. 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 2569 . . . . 5  |-  F/_ x B
4 nfv 1630 . . . . . 6  |-  F/ x  f  Fn  d
5 nfra1 2762 . . . . . 6  |-  F/ x A. x  e.  d 
( f `  x
)  =  ( G `
 Y )
64, 5nfan 1848 . . . . 5  |-  F/ x
( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
73, 6nfrex 2767 . . . 4  |-  F/ x E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
87nfab 2582 . . 3  |-  F/_ x { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
91, 8nfcxfr 2575 . 2  |-  F/_ x C
109nfcrii 2571 1  |-  ( w  e.  C  ->  A. x  w  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1727   {cab 2428   A.wral 2711   E.wrex 2712    Fn wfn 5478   ` cfv 5483
This theorem is referenced by:  bnj1311  29491  bnj1373  29497  bnj1498  29528  bnj1525  29536  bnj1523  29538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717
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