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Theorem bnj1520 28607
Description: Technical lemma for bnj1500 28609. 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 28365 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
43nfcii 2443 . . . . . 6  |-  F/_ f C
54nfuni 3870 . . . . 5  |-  F/_ f U. C
61, 5nfcxfr 2449 . . . 4  |-  F/_ f F
7 nfcv 2452 . . . 4  |-  F/_ f
x
86, 7nffv 5570 . . 3  |-  F/_ f
( F `  x
)
9 nfcv 2452 . . . 4  |-  F/_ f G
10 nfcv 2452 . . . . . 6  |-  F/_ f  pred ( x ,  A ,  R )
116, 10nfres 4994 . . . . 5  |-  F/_ f
( F  |`  pred (
x ,  A ,  R ) )
127, 11nfop 3849 . . . 4  |-  F/_ f <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
139, 12nffv 5570 . . 3  |-  F/_ f
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
148, 13nfeq 2459 . 2  |-  F/ f ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1514nfri 1766 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 set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1531    = wceq 1633   {cab 2302   A.wral 2577   E.wrex 2578    C_ wss 3186   <.cop 3677   U.cuni 3864    |` cres 4728    Fn wfn 5287   ` cfv 5292    predc-bnj14 28224
This theorem is referenced by:  bnj1501  28608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-res 4738  df-iota 5256  df-fv 5300
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