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Theorem erdszelem3 24659
Description: Lemma for erdsze 24668. (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 6029 . . . . . 6  |-  ( x  =  A  ->  (
1 ... x )  =  ( 1 ... A
) )
21pweqd 3748 . . . . 5  |-  ( x  =  A  ->  ~P ( 1 ... x
)  =  ~P (
1 ... A ) )
3 eleq1 2448 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  y  <->  A  e.  y ) )
43anbi2d 685 . . . . 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 2895 . . . 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 5144 . . 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 7387 . 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 9090 . . 3  |-  <  Or  RR
109supex 7402 . 2  |-  sup (
( # " { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) ,  RR ,  <  )  e.  _V
117, 8, 10fvmpt 5746 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 359    = wceq 1649    e. wcel 1717   {crab 2654   ~Pcpw 3743    e. cmpt 4208    |` cres 4821   "cima 4822   -1-1->wf1 5392   ` cfv 5395    Isom wiso 5396  (class class class)co 6021   supcsup 7381   RRcr 8923   1c1 8925    < clt 9054   NNcn 9933   ...cfz 10976   #chash 11546
This theorem is referenced by:  erdszelem5  24661  erdszelem8  24664
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-pre-lttri 8998  ax-pre-lttrn 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-ltxr 9059
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