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Theorem bnj1518 29323
Description: Technical lemma for bnj1500 29327. 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 1726 . . . 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 2862 . . . . . . 7  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
54nfab 2566 . . . . . 6  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
63, 5nfcxfr 2560 . . . . 5  |-  F/_ d C
76nfcri 2555 . . . 4  |-  F/ d  f  e.  C
8 nfv 1726 . . . 4  |-  F/ d  x  e.  dom  f
92, 7, 8nf3an 1956 . . 3  |-  F/ d ( ph  /\  f  e.  C  /\  x  e.  dom  f )
101, 9nfxfr 1664 . 2  |-  F/ d ps
1110nfri 1896 1  |-  ( ps 
->  A. d ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972   A.wal 1401    = wceq 1403    e. wcel 1840   {cab 2385   A.wral 2751   E.wrex 2752    C_ wss 3411   <.cop 3975   U.cuni 4188   dom cdm 4940    |` cres 4942    Fn wfn 5518   ` cfv 5523    predc-bnj14 28943    FrSe w-bnj15 28947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-rex 2757
This theorem is referenced by:  bnj1501  29326
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