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Theorem bnj553 28607
Description: Technical lemma for bnj852 28630. 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
bnj553.1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj553.2  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj553.3  |-  D  =  ( om  \  { (/)
} )
bnj553.4  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj553.5  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj553.6  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj553.7  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
bnj553.8  |-  G  =  ( f  u.  { <. m ,  C >. } )
bnj553.9  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
bnj553.10  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
bnj553.11  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
bnj553.12  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Assertion
Ref Expression
bnj553  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  ( G `  m )  =  L )
Distinct variable groups:    A, i, p, y    y, G    R, i, p, y    f, i, p, y    i, m, p    p, ph'
Allowed substitution hints:    ta( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    A( x, f, m, n)    B( x, y, f, i, m, n, p)    C( x, y, f, i, m, n, p)    D( x, y, f, i, m, n, p)    R( x, f, m, n)    G( x, f, i, m, n, p)    K( x, y, f, i, m, n, p)    L( x, y, f, i, m, n, p)    ph'( x, y, f, i, m, n)    ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj553
StepHypRef Expression
1 bnj553.12 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
21bnj930 28478 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
3 opex 4368 . . . . . . 7  |-  <. m ,  C >.  e.  _V
43snid 3784 . . . . . 6  |-  <. m ,  C >.  e.  { <. m ,  C >. }
5 elun2 3458 . . . . . 6  |-  ( <.
m ,  C >.  e. 
{ <. m ,  C >. }  ->  <. m ,  C >.  e.  (
f  u.  { <. m ,  C >. } ) )
64, 5ax-mp 8 . . . . 5  |-  <. m ,  C >.  e.  (
f  u.  { <. m ,  C >. } )
7 bnj553.8 . . . . 5  |-  G  =  ( f  u.  { <. m ,  C >. } )
86, 7eleqtrri 2460 . . . 4  |-  <. m ,  C >.  e.  G
9 funopfv 5705 . . . 4  |-  ( Fun 
G  ->  ( <. m ,  C >.  e.  G  ->  ( G `  m
)  =  C ) )
102, 8, 9ee10 1382 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( G `  m
)  =  C )
11103ad2ant1 978 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  ( G `  m )  =  C )
12 fveq2 5668 . . . . . 6  |-  ( p  =  i  ->  ( G `  p )  =  ( G `  i ) )
1312bnj1113 28494 . . . . 5  |-  ( p  =  i  ->  U_ y  e.  ( G `  p
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
14 bnj553.11 . . . . 5  |-  L  = 
U_ y  e.  ( G `  p ) 
pred ( y ,  A ,  R )
15 bnj553.10 . . . . 5  |-  K  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R )
1613, 14, 153eqtr4g 2444 . . . 4  |-  ( p  =  i  ->  L  =  K )
17163ad2ant3 980 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  L  =  K )
18 bnj553.5 . . . . 5  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
19 bnj553.9 . . . . 5  |-  B  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R )
20 bnj553.4 . . . . 5  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
2118, 19, 15, 20, 1bnj548 28606 . . . 4  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m
)  ->  B  =  K )
22213adant3 977 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  B  =  K )
23 fveq2 5668 . . . . . 6  |-  ( p  =  i  ->  (
f `  p )  =  ( f `  i ) )
2423bnj1113 28494 . . . . 5  |-  ( p  =  i  ->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
25 bnj553.7 . . . . . . 7  |-  C  = 
U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )
2619, 25eqeq12i 2400 . . . . . 6  |-  ( B  =  C  <->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R ) )
27 eqcom 2389 . . . . . 6  |-  ( U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  =  U_ y  e.  ( f `  p )  pred (
y ,  A ,  R )  <->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2826, 27bitri 241 . . . . 5  |-  ( B  =  C  <->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2924, 28sylibr 204 . . . 4  |-  ( p  =  i  ->  B  =  C )
30293ad2ant3 980 . . 3  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  B  =  C )
3117, 22, 303eqtr2rd 2426 . 2  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  C  =  L )
3211, 31eqtrd 2419 1  |-  ( ( ( R  FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i
)  ->  ( G `  m )  =  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649    \ cdif 3260    u. cun 3261   (/)c0 3571   {csn 3757   <.cop 3760   U_ciun 4035   suc csuc 4524   omcom 4785   Fun wfun 5388    Fn wfn 5389   ` cfv 5394    predc-bnj14 28390    FrSe w-bnj15 28394
This theorem is referenced by:  bnj557  28610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402
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