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Theorem bnj548 32245
Description: Technical lemma for bnj852 32269. 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 32118 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
32adantr 465 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  Fun  G )
4 bnj548.1 . . . . . . . 8  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
54simp1bi 1003 . . . . . . 7  |-  ( ta 
->  f  Fn  m
)
6 fndm 5621 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
7 eleq2 2527 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( i  e.  dom  f 
<->  i  e.  m ) )
87biimpar 485 . . . . . . . 8  |-  ( ( dom  f  =  m  /\  i  e.  m
)  ->  i  e.  dom  f )
96, 8sylan 471 . . . . . . 7  |-  ( ( f  Fn  m  /\  i  e.  m )  ->  i  e.  dom  f
)
105, 9sylan 471 . . . . . 6  |-  ( ( ta  /\  i  e.  m )  ->  i  e.  dom  f )
11103ad2antl2 1151 . . . . 5  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  i  e.  dom  f )
123, 11jca 532 . . . 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 32119 . . . 4  |-  f  C_  G
1512, 14jctil 537 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
16 3anan12 978 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  <->  ( f  C_  G  /\  ( Fun 
G  /\  i  e.  dom  f ) ) )
1715, 16sylibr 212 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  ( Fun  G  /\  f  C_  G  /\  i  e.  dom  f ) )
18 funssfv 5817 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  i  e. 
dom  f )  -> 
( G `  i
)  =  ( f `
 i ) )
19 iuneq1 4295 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2019eqcomd 2462 . . 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 2520 . 2  |-  ( ( G `  i )  =  ( f `  i )  ->  B  =  K )
2417, 18, 233syl 20 1  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  B  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    u. cun 3437    C_ wss 3439   {csn 3988   <.cop 3994   U_ciun 4282   dom cdm 4951   Fun wfun 5523    Fn wfn 5524   ` cfv 5529    predc-bnj14 32031    FrSe w-bnj15 32035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537
This theorem is referenced by:  bnj553  32246
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