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Theorem erdszelem3 24871
Description: Lemma for erdsze 24880. (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 6081 . . . . . 6  |-  ( x  =  A  ->  (
1 ... x )  =  ( 1 ... A
) )
21pweqd 3796 . . . . 5  |-  ( x  =  A  ->  ~P ( 1 ... x
)  =  ~P (
1 ... A ) )
3 eleq1 2495 . . . . . 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 2943 . . . 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 5195 . . 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 7443 . 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 9148 . . 3  |-  <  Or  RR
109supex 7460 . 2  |-  sup (
( # " { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y ) } ) ,  RR ,  <  )  e.  _V
117, 8, 10fvmpt 5798 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 1652    e. wcel 1725   {crab 2701   ~Pcpw 3791    e. cmpt 4258    |` cres 4872   "cima 4873   -1-1->wf1 5443   ` cfv 5446    Isom wiso 5447  (class class class)co 6073   supcsup 7437   RRcr 8981   1c1 8983    < clt 9112   NNcn 9992   ...cfz 11035   #chash 11610
This theorem is referenced by:  erdszelem5  24873  erdszelem8  24876
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-ltxr 9117
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