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Theorem bnj958 28972
Description: Technical lemma for bnj69 29040. 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 2419 . . . . . 6  |-  F/_ y
f
3 nfcv 2419 . . . . . . . 8  |-  F/_ y
n
4 bnj958.1 . . . . . . . . 9  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
5 nfiu1 3933 . . . . . . . . 9  |-  F/_ y U_ y  e.  (
f `  m )  pred ( y ,  A ,  R )
64, 5nfcxfr 2416 . . . . . . . 8  |-  F/_ y C
73, 6nfop 3812 . . . . . . 7  |-  F/_ y <. n ,  C >.
87nfsn 3691 . . . . . 6  |-  F/_ y { <. n ,  C >. }
92, 8nfun 3331 . . . . 5  |-  F/_ y
( f  u.  { <. n ,  C >. } )
101, 9nfcxfr 2416 . . . 4  |-  F/_ y G
11 nfcv 2419 . . . 4  |-  F/_ y
i
1210, 11nffv 5532 . . 3  |-  F/_ y
( G `  i
)
1312nfeq1 2428 . 2  |-  F/ y ( G `  i
)  =  ( f `
 i )
1413nfri 1742 1  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    u. cun 3150   {csn 3640   <.cop 3643   U_ciun 3905   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj966  28976  bnj967  28977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-iota 5219  df-fv 5263
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