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Theorem bnj1520 32054
Description: Technical lemma for bnj1500 32056. 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
bnj1520.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1520.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1520.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1520.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj1520  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
Distinct variable groups:    A, f    f, G    R, f    x, f
Allowed substitution hints:    A( x, d)    B( x, f, d)    C( x, f, d)    R( x, d)    F( x, f, d)    G( x, d)    Y( x, f, d)

Proof of Theorem bnj1520
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1520.4 . . . . 5  |-  F  = 
U. C
2 bnj1520.3 . . . . . . . 8  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
32bnj1317 31812 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
43nfcii 2568 . . . . . 6  |-  F/_ f C
54nfuni 4095 . . . . 5  |-  F/_ f U. C
61, 5nfcxfr 2574 . . . 4  |-  F/_ f F
7 nfcv 2577 . . . 4  |-  F/_ f
x
86, 7nffv 5696 . . 3  |-  F/_ f
( F `  x
)
9 nfcv 2577 . . . 4  |-  F/_ f G
10 nfcv 2577 . . . . . 6  |-  F/_ f  pred ( x ,  A ,  R )
116, 10nfres 5110 . . . . 5  |-  F/_ f
( F  |`  pred (
x ,  A ,  R ) )
127, 11nfop 4073 . . . 4  |-  F/_ f <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
139, 12nffv 5696 . . 3  |-  F/_ f
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
148, 13nfeq 2584 . 2  |-  F/ f ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1514nfri 1808 1  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367    = wceq 1369   {cab 2427   A.wral 2713   E.wrex 2714    C_ wss 3326   <.cop 3881   U.cuni 4089    |` cres 4840    Fn wfn 5411   ` cfv 5416    predc-bnj14 31673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-xp 4844  df-res 4850  df-iota 5379  df-fv 5424
This theorem is referenced by:  bnj1501  32055
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