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Theorem bnj1518 29875
Description: Technical lemma for bnj1500 29879. 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
bnj1518.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1518.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1518.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1518.4  |-  F  = 
U. C
bnj1518.5  |-  ( ph  <->  ( R  FrSe  A  /\  x  e.  A )
)
bnj1518.6  |-  ( ps  <->  (
ph  /\  f  e.  C  /\  x  e.  dom  f ) )
Assertion
Ref Expression
bnj1518  |-  ( ps 
->  A. d ps )
Distinct variable groups:    f, d    ph, d    x, d
Allowed substitution hints:    ph( x, f)    ps( x, f, d)    A( x, f, d)    B( x, f, d)    C( x, f, d)    R( x, f, d)    F( x, f, d)    G( x, f, d)    Y( x, f, d)

Proof of Theorem bnj1518
StepHypRef Expression
1 bnj1518.6 . . 3  |-  ( ps  <->  (
ph  /\  f  e.  C  /\  x  e.  dom  f ) )
2 nfv 1752 . . . 4  |-  F/ d
ph
3 bnj1518.3 . . . . . 6  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 nfre1 2887 . . . . . . 7  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
54nfab 2589 . . . . . 6  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
63, 5nfcxfr 2583 . . . . 5  |-  F/_ d C
76nfcri 2578 . . . 4  |-  F/ d  f  e.  C
8 nfv 1752 . . . 4  |-  F/ d  x  e.  dom  f
92, 7, 8nf3an 1987 . . 3  |-  F/ d ( ph  /\  f  e.  C  /\  x  e.  dom  f )
101, 9nfxfr 1693 . 2  |-  F/ d ps
1110nfri 1926 1  |-  ( ps 
->  A. d ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983   A.wal 1436    = wceq 1438    e. wcel 1869   {cab 2408   A.wral 2776   E.wrex 2777    C_ wss 3437   <.cop 4003   U.cuni 4217   dom cdm 4851    |` cres 4853    Fn wfn 5594   ` cfv 5599    predc-bnj14 29495    FrSe w-bnj15 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rex 2782
This theorem is referenced by:  bnj1501  29878
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