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Theorem bnj1520 34523
Description: Technical lemma for bnj1500 34525. 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 34281 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
43nfcii 2606 . . . . . 6  |-  F/_ f C
54nfuni 4241 . . . . 5  |-  F/_ f U. C
61, 5nfcxfr 2614 . . . 4  |-  F/_ f F
7 nfcv 2616 . . . 4  |-  F/_ f
x
86, 7nffv 5855 . . 3  |-  F/_ f
( F `  x
)
9 nfcv 2616 . . . 4  |-  F/_ f G
10 nfcv 2616 . . . . . 6  |-  F/_ f  pred ( x ,  A ,  R )
116, 10nfres 5264 . . . . 5  |-  F/_ f
( F  |`  pred (
x ,  A ,  R ) )
127, 11nfop 4219 . . . 4  |-  F/_ f <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
139, 12nffv 5855 . . 3  |-  F/_ f
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
148, 13nfeq 2627 . 2  |-  F/ f ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1514nfri 1879 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 367   A.wal 1396    = wceq 1398   {cab 2439   A.wral 2804   E.wrex 2805    C_ wss 3461   <.cop 4022   U.cuni 4235    |` cres 4990    Fn wfn 5565   ` cfv 5570    predc-bnj14 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-res 5000  df-iota 5534  df-fv 5578
This theorem is referenced by:  bnj1501  34524
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