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Theorem bnj958 28642
Description: Technical lemma for bnj69 28710. 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 2516 . . . . . 6  |-  F/_ y
f
3 nfcv 2516 . . . . . . . 8  |-  F/_ y
n
4 bnj958.1 . . . . . . . . 9  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
5 nfiu1 4056 . . . . . . . . 9  |-  F/_ y U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )
64, 5nfcxfr 2513 . . . . . . . 8  |-  F/_ y C
73, 6nfop 3935 . . . . . . 7  |-  F/_ y <. n ,  C >.
87nfsn 3802 . . . . . 6  |-  F/_ y { <. n ,  C >. }
92, 8nfun 3439 . . . . 5  |-  F/_ y
( f  u.  { <. n ,  C >. } )
101, 9nfcxfr 2513 . . . 4  |-  F/_ y G
11 nfcv 2516 . . . 4  |-  F/_ y
i
1210, 11nffv 5668 . . 3  |-  F/_ y
( G `  i
)
1312nfeq1 2525 . 2  |-  F/ y ( G `  i
)  =  ( f `
 i )
1413nfri 1770 1  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    = wceq 1649    u. cun 3254   {csn 3750   <.cop 3753   U_ciun 4028   ` cfv 5387    predc-bnj14 28383
This theorem is referenced by:  bnj966  28646  bnj967  28647
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-iota 5351  df-fv 5395
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