Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem3 Unicode version

Theorem erdszelem3 23739
Description: Lemma for erdsze 23748. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.k  |-  K  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
Assertion
Ref Expression
erdszelem3  |-  ( A  e.  ( 1 ... N )  ->  ( K `  A )  =  sup ( ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) ,  RR ,  <  )
)
Distinct variable groups:    x, y, F    x, A, y    x, O, y    x, N, y    ph, x, y
Allowed substitution hints:    K( x, y)

Proof of Theorem erdszelem3
StepHypRef Expression
1 oveq2 5882 . . . . . 6  |-  ( x  =  A  ->  (
1 ... x )  =  ( 1 ... A
) )
21pweqd 3643 . . . . 5  |-  ( x  =  A  ->  ~P ( 1 ... x
)  =  ~P (
1 ... A ) )
3 eleq1 2356 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  y  <->  A  e.  y ) )
43anbi2d 684 . . . . 5  |-  ( x  =  A  ->  (
( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  x  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F "
y ) )  /\  A  e.  y )
) )
52, 4rabeqbidv 2796 . . . 4  |-  ( x  =  A  ->  { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  x  e.  y ) }  =  { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } )
65imaeq2d 5028 . . 3  |-  ( x  =  A  ->  ( #
" { y  e. 
~P ( 1 ... x )  |  ( ( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  /\  x  e.  y ) } )  =  ( # " {
y  e.  ~P (
1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) )
76supeq1d 7215 . 2  |-  ( x  =  A  ->  sup ( ( # " {
y  e.  ~P (
1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) ,  RR ,  <  )  =  sup ( (
# " { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) ,  RR ,  <  ) )
8 erdszelem.k . 2  |-  K  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
9 ltso 8919 . . 3  |-  <  Or  RR
109supex 7230 . 2  |-  sup (
( # " { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) ,  RR ,  <  )  e.  _V
117, 8, 10fvmpt 5618 1  |-  ( A  e.  ( 1 ... N )  ->  ( K `  A )  =  sup ( ( # " { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) } ) ,  RR ,  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   ~Pcpw 3638    e. cmpt 4093    |` cres 4707   "cima 4708   -1-1->wf1 5268   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883   NNcn 9762   ...cfz 10798   #chash 11353
This theorem is referenced by:  erdszelem5  23741  erdszelem8  23744
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-ltxr 8888
  Copyright terms: Public domain W3C validator