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Theorem bnj548 28302
Description: Technical lemma for bnj852 28326. 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 28174 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
32adantr 451 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  Fun  G )
4 bnj548.1 . . . . . . . 8  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
54simp1bi 970 . . . . . . 7  |-  ( ta 
->  f  Fn  m
)
6 fndm 5343 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
7 eleq2 2344 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
87biimpar 471 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
96, 8sylan 457 . . . . . . 7  |-  ( ( f  Fn  m  /\  i  e.  m )  ->  i  e.  dom  f
)
105, 9sylan 457 . . . . . 6  |-  ( ( ta  /\  i  e.  m )  ->  i  e.  dom  f )
11103ad2antl2 1118 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  i  e.  dom  f )
123, 11jca 518 . . . 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 28175 . . . 4  |-  f  C_  G
1512, 14jctil 523 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
16 3anan12 947 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  <->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
1715, 16sylibr 203 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( Fun  G  /\  f  C_  G  /\  i  e.  dom  f ) )
18 funssfv 5543 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  -> 
( G `  i
)  =  ( f `
 i ) )
19 iuneq1 3918 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2019eqcomd 2288 . . 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 2340 . 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    predc-bnj14 28086    FrSe w-bnj15 28090
This theorem is referenced by:  bnj553  28303
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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