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Theorem rdglim2 6626
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 6620 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. ( rec ( F ,  A ) " B ) )
2 dfima3 5146 . . . . 5  |-  ( rec ( F ,  A
) " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) ) }
3 df-rex 2655 . . . . . . 7  |-  ( E. x  e.  B  y  =  ( rec ( F ,  A ) `  x )  <->  E. x
( x  e.  B  /\  y  =  ( rec ( F ,  A
) `  x )
) )
4 limord 4581 . . . . . . . . . . 11  |-  ( Lim 
B  ->  Ord  B )
5 ordelord 4544 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  x  e.  B )  ->  Ord  x )
65ex 424 . . . . . . . . . . . 12  |-  ( Ord 
B  ->  ( x  e.  B  ->  Ord  x
) )
7 vex 2902 . . . . . . . . . . . . 13  |-  x  e. 
_V
87elon 4531 . . . . . . . . . . . 12  |-  ( x  e.  On  <->  Ord  x )
96, 8syl6ibr 219 . . . . . . . . . . 11  |-  ( Ord 
B  ->  ( x  e.  B  ->  x  e.  On ) )
104, 9syl 16 . . . . . . . . . 10  |-  ( Lim 
B  ->  ( x  e.  B  ->  x  e.  On ) )
11 eqcom 2389 . . . . . . . . . . 11  |-  ( y  =  ( rec ( F ,  A ) `  x )  <->  ( rec ( F ,  A ) `
 x )  =  y )
12 rdgfnon 6612 . . . . . . . . . . . 12  |-  rec ( F ,  A )  Fn  On
13 fnopfvb 5707 . . . . . . . . . . . 12  |-  ( ( rec ( F ,  A )  Fn  On  /\  x  e.  On )  ->  ( ( rec ( F ,  A
) `  x )  =  y  <->  <. x ,  y
>.  e.  rec ( F ,  A ) ) )
1412, 13mpan 652 . . . . . . . . . . 11  |-  ( x  e.  On  ->  (
( rec ( F ,  A ) `  x )  =  y  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) )
1511, 14syl5bb 249 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
y  =  ( rec ( F ,  A
) `  x )  <->  <.
x ,  y >.  e.  rec ( F ,  A ) ) )
1610, 15syl6 31 . . . . . . . . 9  |-  ( Lim 
B  ->  ( x  e.  B  ->  ( y  =  ( rec ( F ,  A ) `  x )  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) ) )
1716pm5.32d 621 . . . . . . . 8  |-  ( Lim 
B  ->  ( (
x  e.  B  /\  y  =  ( rec ( F ,  A ) `
 x ) )  <-> 
( x  e.  B  /\  <. x ,  y
>.  e.  rec ( F ,  A ) ) ) )
1817exbidv 1633 . . . . . . 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 250 . . . . . 6  |-  ( Lim 
B  ->  ( E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) )  <->  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) ) )
2019abbidv 2501 . . . . 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 2431 . . . 4  |-  ( Lim 
B  ->  ( rec ( F ,  A )
" B )  =  { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x ) } )
2221unieqd 3968 . . 3  |-  ( Lim 
B  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
2322adantl 453 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
241, 23eqtrd 2419 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   E.wrex 2650   <.cop 3760   U.cuni 3957   Ord word 4521   Oncon0 4522   Lim wlim 4523   "cima 4821    Fn wfn 5389   ` cfv 5394   reccrdg 6603
This theorem is referenced by:  rdglim2a  6627
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 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-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-recs 6569  df-rdg 6604
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