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Theorem bnj548 29708
Description: Technical lemma for bnj852 29732. 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
bnj548.1  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj548.2  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj548.3  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj548.4  |-  G  =  ( f  u.  { <. m ,  C >. } )
bnj548.5  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Assertion
Ref Expression
bnj548  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  B  =  K )
Distinct variable groups:    y, G    y, f    y, i
Allowed substitution hints:    ta( y, f, i, m, n)    si( y,
f, i, m, n)    A( y, f, i, m, n)    B( y, f, i, m, n)    C( y,
f, i, m, n)    R( y, f, i, m, n)    G( f, i, m, n)    K( y, f, i, m, n)    ph'( y, f, i, m, n)    ps'( y, f, i, m, n)

Proof of Theorem bnj548
StepHypRef Expression
1 bnj548.5 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
21bnj930 29581 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
32adantr 467 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  Fun  G )
4 bnj548.1 . . . . . . . 8  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
54simp1bi 1023 . . . . . . 7  |-  ( ta 
->  f  Fn  m
)
6 fndm 5675 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
7 eleq2 2518 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
87biimpar 488 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
96, 8sylan 474 . . . . . . 7  |-  ( ( f  Fn  m  /\  i  e.  m )  ->  i  e.  dom  f
)
105, 9sylan 474 . . . . . 6  |-  ( ( ta  /\  i  e.  m )  ->  i  e.  dom  f )
11103ad2antl2 1171 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  i  e.  dom  f )
123, 11jca 535 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( Fun  G  /\  i  e.  dom  f ) )
13 bnj548.4 . . . . 5  |-  G  =  ( f  u.  { <. m ,  C >. } )
1413bnj931 29582 . . . 4  |-  f  C_  G
1512, 14jctil 540 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
16 3anan12 998 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  <->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
1715, 16sylibr 216 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( Fun  G  /\  f  C_  G  /\  i  e.  dom  f ) )
18 funssfv 5880 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  -> 
( G `  i
)  =  ( f `
 i ) )
19 iuneq1 4292 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2019eqcomd 2457 . . 3  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
21 bnj548.2 . . 3  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
22 bnj548.3 . . 3  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
2320, 21, 223eqtr4g 2510 . 2  |-  ( ( G `  i )  =  ( f `  i )  ->  B  =  K )
2417, 18, 233syl 18 1  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  B  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    u. cun 3402    C_ wss 3404   {csn 3968   <.cop 3974   U_ciun 4278   dom cdm 4834   Fun wfun 5576    Fn wfn 5577   ` cfv 5582    predc-bnj14 29493    FrSe w-bnj15 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590
This theorem is referenced by:  bnj553  29709
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