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Theorem bnj958 29212
Description: Technical lemma for bnj69 29280. 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
bnj958.1  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj958.2  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj958  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
Distinct variable groups:    y, f    y, i    y, n
Allowed substitution hints:    A( y, f, i, m, n)    C( y, f, i, m, n)    R( y, f, i, m, n)    G( y, f, i, m, n)

Proof of Theorem bnj958
StepHypRef Expression
1 bnj958.2 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
2 nfcv 2571 . . . . . 6  |-  F/_ y
f
3 nfcv 2571 . . . . . . . 8  |-  F/_ y
n
4 bnj958.1 . . . . . . . . 9  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
5 nfiu1 4113 . . . . . . . . 9  |-  F/_ y U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )
64, 5nfcxfr 2568 . . . . . . . 8  |-  F/_ y C
73, 6nfop 3992 . . . . . . 7  |-  F/_ y <. n ,  C >.
87nfsn 3858 . . . . . 6  |-  F/_ y { <. n ,  C >. }
92, 8nfun 3495 . . . . 5  |-  F/_ y
( f  u.  { <. n ,  C >. } )
101, 9nfcxfr 2568 . . . 4  |-  F/_ y G
11 nfcv 2571 . . . 4  |-  F/_ y
i
1210, 11nffv 5727 . . 3  |-  F/_ y
( G `  i
)
1312nfeq1 2580 . 2  |-  F/ y ( G `  i
)  =  ( f `
 i )
1413nfri 1778 1  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652    u. cun 3310   {csn 3806   <.cop 3809   U_ciun 4085   ` cfv 5446    predc-bnj14 28953
This theorem is referenced by:  bnj966  29216  bnj967  29217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-iota 5410  df-fv 5454
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