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Theorem bnj548 29245
Description: Technical lemma for bnj852 29269. 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 29117 . . . . . 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 5359 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
7 eleq2 2357 . . . . . . . . 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 29118 . . . 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 5559 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  -> 
( G `  i
)  =  ( f `
 i ) )
19 iuneq1 3934 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2019eqcomd 2301 . . 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 2353 . 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 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   {csn 3653   <.cop 3656   U_ciun 3921   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj553  29246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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