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Theorem bnj1520 29497
Description: Technical lemma for bnj1500 29499. 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 29255 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
43nfcii 2565 . . . . . 6  |-  F/_ f C
54nfuni 4023 . . . . 5  |-  F/_ f U. C
61, 5nfcxfr 2571 . . . 4  |-  F/_ f F
7 nfcv 2574 . . . 4  |-  F/_ f
x
86, 7nffv 5737 . . 3  |-  F/_ f
( F `  x
)
9 nfcv 2574 . . . 4  |-  F/_ f G
10 nfcv 2574 . . . . . 6  |-  F/_ f  pred ( x ,  A ,  R )
116, 10nfres 5150 . . . . 5  |-  F/_ f
( F  |`  pred (
x ,  A ,  R ) )
127, 11nfop 4002 . . . 4  |-  F/_ f <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
139, 12nffv 5737 . . 3  |-  F/_ f
( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)
148, 13nfeq 2581 . 2  |-  F/ f ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
1514nfri 1779 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 360   A.wal 1550    = wceq 1653   {cab 2424   A.wral 2707   E.wrex 2708    C_ wss 3322   <.cop 3819   U.cuni 4017    |` cres 4882    Fn wfn 5451   ` cfv 5456    predc-bnj14 29114
This theorem is referenced by:  bnj1501  29498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-res 4892  df-iota 5420  df-fv 5464
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