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Theorem bnj1519 29411
Description: Technical lemma for bnj1500 29414. 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
bnj1519.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1519.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1519.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1519.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj1519  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
Distinct variable groups:    A, d    G, d    R, d    x, d
Allowed substitution hints:    A( x, f)    B( x, f, d)    C( x, f, d)    R( x, f)    F( x, f, d)    G( x, f)    Y( x, f, d)

Proof of Theorem bnj1519
StepHypRef Expression
1 bnj1519.4 . . . . 5  |-  F  = 
U. C
2 bnj1519.3 . . . . . . 7  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
3 nfre1 2612 . . . . . . . 8  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
43nfab 2436 . . . . . . 7  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
52, 4nfcxfr 2429 . . . . . 6  |-  F/_ d C
65nfuni 3849 . . . . 5  |-  F/_ d U. C
71, 6nfcxfr 2429 . . . 4  |-  F/_ d F
8 nfcv 2432 . . . 4  |-  F/_ d
x
97, 8nffv 5548 . . 3  |-  F/_ d
( F `  x
)
10 nfcv 2432 . . . 4  |-  F/_ d G
11 nfcv 2432 . . . . . 6  |-  F/_ d  pred ( x ,  A ,  R )
127, 11nfres 4973 . . . . 5  |-  F/_ d
( F  |`  pred (
x ,  A ,  R ) )
138, 12nfop 3828 . . . 4  |-  F/_ d <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
1410, 13nffv 5548 . . 3  |-  F/_ d
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
159, 14nfeq 2439 . 2  |-  F/ d ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1615nfri 1754 1  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   <.cop 3656   U.cuni 3843    |` cres 4707    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj1501  29413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-res 4717  df-iota 5235  df-fv 5279
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