MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rdglim2 Structured version   Unicode version

Theorem rdglim2 7161
Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)
Assertion
Ref Expression
rdglim2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) } )
Distinct variable groups:    x, y, A    x, B, y    x, F, y
Allowed substitution hints:    C( x, y)

Proof of Theorem rdglim2
StepHypRef Expression
1 rdglim 7155 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. ( rec ( F ,  A ) " B ) )
2 dfima3 5190 . . . . 5  |-  ( rec ( F ,  A
) " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) ) }
3 df-rex 2777 . . . . . . 7  |-  ( E. x  e.  B  y  =  ( rec ( F ,  A ) `  x )  <->  E. x
( x  e.  B  /\  y  =  ( rec ( F ,  A
) `  x )
) )
4 limord 5501 . . . . . . . . . . 11  |-  ( Lim 
B  ->  Ord  B )
5 ordelord 5464 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  x  e.  B )  ->  Ord  x )
65ex 435 . . . . . . . . . . . 12  |-  ( Ord 
B  ->  ( x  e.  B  ->  Ord  x
) )
7 vex 3083 . . . . . . . . . . . . 13  |-  x  e. 
_V
87elon 5451 . . . . . . . . . . . 12  |-  ( x  e.  On  <->  Ord  x )
96, 8syl6ibr 230 . . . . . . . . . . 11  |-  ( Ord 
B  ->  ( x  e.  B  ->  x  e.  On ) )
104, 9syl 17 . . . . . . . . . 10  |-  ( Lim 
B  ->  ( x  e.  B  ->  x  e.  On ) )
11 eqcom 2431 . . . . . . . . . . 11  |-  ( y  =  ( rec ( F ,  A ) `  x )  <->  ( rec ( F ,  A ) `
 x )  =  y )
12 rdgfnon 7147 . . . . . . . . . . . 12  |-  rec ( F ,  A )  Fn  On
13 fnopfvb 5922 . . . . . . . . . . . 12  |-  ( ( rec ( F ,  A )  Fn  On  /\  x  e.  On )  ->  ( ( rec ( F ,  A
) `  x )  =  y  <->  <. x ,  y
>.  e.  rec ( F ,  A ) ) )
1412, 13mpan 674 . . . . . . . . . . 11  |-  ( x  e.  On  ->  (
( rec ( F ,  A ) `  x )  =  y  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) )
1511, 14syl5bb 260 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
y  =  ( rec ( F ,  A
) `  x )  <->  <.
x ,  y >.  e.  rec ( F ,  A ) ) )
1610, 15syl6 34 . . . . . . . . 9  |-  ( Lim 
B  ->  ( x  e.  B  ->  ( y  =  ( rec ( F ,  A ) `  x )  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) ) )
1716pm5.32d 643 . . . . . . . 8  |-  ( Lim 
B  ->  ( (
x  e.  B  /\  y  =  ( rec ( F ,  A ) `
 x ) )  <-> 
( x  e.  B  /\  <. x ,  y
>.  e.  rec ( F ,  A ) ) ) )
1817exbidv 1762 . . . . . . 7  |-  ( Lim 
B  ->  ( E. x ( x  e.  B  /\  y  =  ( rec ( F ,  A ) `  x ) )  <->  E. x
( x  e.  B  /\  <. x ,  y
>.  e.  rec ( F ,  A ) ) ) )
193, 18syl5rbb 261 . . . . . 6  |-  ( Lim 
B  ->  ( E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) )  <->  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) ) )
2019abbidv 2553 . . . . 5  |-  ( Lim 
B  ->  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) ) }  =  {
y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
212, 20syl5eq 2475 . . . 4  |-  ( Lim 
B  ->  ( rec ( F ,  A )
" B )  =  { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x ) } )
2221unieqd 4229 . . 3  |-  ( Lim 
B  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
2322adantl 467 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
241, 23eqtrd 2463 1  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407   E.wrex 2772   <.cop 4004   U.cuni 4219   "cima 4856   Ord word 5441   Oncon0 5442   Lim wlim 5443    Fn wfn 5596   ` cfv 5601   reccrdg 7138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-wrecs 7039  df-recs 7101  df-rdg 7139
This theorem is referenced by:  rdglim2a  7162
  Copyright terms: Public domain W3C validator