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Theorem bnj1371 29059
Description: Technical lemma for bnj60 29092. 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
bnj1371.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1371.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1371.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1371.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1371.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1371.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1371.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1371.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1371.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1371.10  |-  P  = 
U. H
bnj1371.11  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
Assertion
Ref Expression
bnj1371  |-  ( f  e.  H  ->  Fun  f )
Distinct variable groups:    f, d    y, f
Allowed substitution hints:    ps( x, y, f, d)    ch( x, y, f, d)    ta( x, y, f, d)    A( x, y, f, d)    B( x, y, f, d)    C( x, y, f, d)    D( x, y, f, d)    P( x, y, f, d)    R( x, y, f, d)    G( x, y, f, d)    H( x, y, f, d)    Y( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1371
StepHypRef Expression
1 bnj1371.9 . . . . . . 7  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
21bnj1436 28872 . . . . . 6  |-  ( f  e.  H  ->  E. y  e.  pred  ( x ,  A ,  R ) ta' )
3 rexex 2602 . . . . . 6  |-  ( E. y  e.  pred  (
x ,  A ,  R ) ta'  ->  E. y ta' )
42, 3syl 15 . . . . 5  |-  ( f  e.  H  ->  E. y ta' )
5 bnj1371.11 . . . . . 6  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
65exbii 1569 . . . . 5  |-  ( E. y ta'  <->  E. y ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
74, 6sylib 188 . . . 4  |-  ( f  e.  H  ->  E. y
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
8 exsimpl 1579 . . . 4  |-  ( E. y ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) )  ->  E. y  f  e.  C )
97, 8syl 15 . . 3  |-  ( f  e.  H  ->  E. y 
f  e.  C )
10 bnj1371.3 . . . . . . 7  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
1110abeq2i 2390 . . . . . 6  |-  ( f  e.  C  <->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
1211bnj1238 28839 . . . . 5  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
13 fnfun 5341 . . . . 5  |-  ( f  Fn  d  ->  Fun  f )
1412, 13bnj31 28745 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  Fun  f )
1514bnj1265 28845 . . 3  |-  ( f  e.  C  ->  Fun  f )
169, 15bnj593 28774 . 2  |-  ( f  e.  H  ->  E. y Fun  f )
1716bnj937 28803 1  |-  ( f  e.  H  ->  Fun  f )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   [.wsbc 2991    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1384  29062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rex 2549  df-fn 5258
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