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Theorem bnj1519 28774
Description: Technical lemma for bnj1500 28777. 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 2707 . . . . . . . 8  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
43nfab 2529 . . . . . . 7  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
52, 4nfcxfr 2522 . . . . . 6  |-  F/_ d C
65nfuni 3965 . . . . 5  |-  F/_ d U. C
71, 6nfcxfr 2522 . . . 4  |-  F/_ d F
8 nfcv 2525 . . . 4  |-  F/_ d
x
97, 8nffv 5677 . . 3  |-  F/_ d
( F `  x
)
10 nfcv 2525 . . . 4  |-  F/_ d G
11 nfcv 2525 . . . . . 6  |-  F/_ d  pred ( x ,  A ,  R )
127, 11nfres 5090 . . . . 5  |-  F/_ d
( F  |`  pred (
x ,  A ,  R ) )
138, 12nfop 3944 . . . 4  |-  F/_ d <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
1410, 13nffv 5677 . . 3  |-  F/_ d
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
159, 14nfeq 2532 . 2  |-  F/ d ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1615nfri 1770 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 359   A.wal 1546    = wceq 1649   {cab 2375   A.wral 2651   E.wrex 2652    C_ wss 3265   <.cop 3762   U.cuni 3959    |` cres 4822    Fn wfn 5391   ` cfv 5396    predc-bnj14 28392
This theorem is referenced by:  bnj1501  28776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-xp 4826  df-res 4832  df-iota 5360  df-fv 5404
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