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Theorem bnj1519 33075
Description: Technical lemma for bnj1500 33078. 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 2918 . . . . . . . 8  |-  F/ d E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )
43nfab 2626 . . . . . . 7  |-  F/_ d { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
52, 4nfcxfr 2620 . . . . . 6  |-  F/_ d C
65nfuni 4244 . . . . 5  |-  F/_ d U. C
71, 6nfcxfr 2620 . . . 4  |-  F/_ d F
8 nfcv 2622 . . . 4  |-  F/_ d
x
97, 8nffv 5864 . . 3  |-  F/_ d
( F `  x
)
10 nfcv 2622 . . . 4  |-  F/_ d G
11 nfcv 2622 . . . . . 6  |-  F/_ d  pred ( x ,  A ,  R )
127, 11nfres 5266 . . . . 5  |-  F/_ d
( F  |`  pred (
x ,  A ,  R ) )
138, 12nfop 4222 . . . 4  |-  F/_ d <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
1410, 13nffv 5864 . . 3  |-  F/_ d
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
159, 14nfeq 2633 . 2  |-  F/ d ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1615nfri 1817 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1372    = wceq 1374   {cab 2445   A.wral 2807   E.wrex 2808    C_ wss 3469   <.cop 4026   U.cuni 4238    |` cres 4994    Fn wfn 5574   ` cfv 5579    predc-bnj14 32695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-res 5004  df-iota 5542  df-fv 5587
This theorem is referenced by:  bnj1501  33077
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