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Theorem bnj548 29206
Description: Technical lemma for bnj852 29230. 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 29078 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
32adantr 452 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  Fun  G )
4 bnj548.1 . . . . . . . 8  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
54simp1bi 972 . . . . . . 7  |-  ( ta 
->  f  Fn  m
)
6 fndm 5537 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
7 eleq2 2497 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
87biimpar 472 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
96, 8sylan 458 . . . . . . 7  |-  ( ( f  Fn  m  /\  i  e.  m )  ->  i  e.  dom  f
)
105, 9sylan 458 . . . . . 6  |-  ( ( ta  /\  i  e.  m )  ->  i  e.  dom  f )
11103ad2antl2 1120 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  i  e.  dom  f )
123, 11jca 519 . . . 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 29079 . . . 4  |-  f  C_  G
1512, 14jctil 524 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
16 3anan12 949 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  <->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
1715, 16sylibr 204 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( Fun  G  /\  f  C_  G  /\  i  e.  dom  f ) )
18 funssfv 5739 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  -> 
( G `  i
)  =  ( f `
 i ) )
19 iuneq1 4099 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2019eqcomd 2441 . . 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 2493 . 2  |-  ( ( G `  i )  =  ( f `  i )  ->  B  =  K )
2417, 18, 233syl 19 1  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  B  =  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    u. cun 3311    C_ wss 3313   {csn 3807   <.cop 3810   U_ciun 4086   dom cdm 4871   Fun wfun 5441    Fn wfn 5442   ` cfv 5447    predc-bnj14 28990    FrSe w-bnj15 28994
This theorem is referenced by:  bnj553  29207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-res 4883  df-iota 5411  df-fun 5449  df-fn 5450  df-fv 5455
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