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Theorem rdglim2 6880
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 6874 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. ( rec ( F ,  A ) " B ) )
2 dfima3 5165 . . . . 5  |-  ( rec ( F ,  A
) " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) ) }
3 df-rex 2715 . . . . . . 7  |-  ( E. x  e.  B  y  =  ( rec ( F ,  A ) `  x )  <->  E. x
( x  e.  B  /\  y  =  ( rec ( F ,  A
) `  x )
) )
4 limord 4770 . . . . . . . . . . 11  |-  ( Lim 
B  ->  Ord  B )
5 ordelord 4733 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  x  e.  B )  ->  Ord  x )
65ex 434 . . . . . . . . . . . 12  |-  ( Ord 
B  ->  ( x  e.  B  ->  Ord  x
) )
7 vex 2969 . . . . . . . . . . . . 13  |-  x  e. 
_V
87elon 4720 . . . . . . . . . . . 12  |-  ( x  e.  On  <->  Ord  x )
96, 8syl6ibr 227 . . . . . . . . . . 11  |-  ( Ord 
B  ->  ( x  e.  B  ->  x  e.  On ) )
104, 9syl 16 . . . . . . . . . 10  |-  ( Lim 
B  ->  ( x  e.  B  ->  x  e.  On ) )
11 eqcom 2439 . . . . . . . . . . 11  |-  ( y  =  ( rec ( F ,  A ) `  x )  <->  ( rec ( F ,  A ) `
 x )  =  y )
12 rdgfnon 6866 . . . . . . . . . . . 12  |-  rec ( F ,  A )  Fn  On
13 fnopfvb 5726 . . . . . . . . . . . 12  |-  ( ( rec ( F ,  A )  Fn  On  /\  x  e.  On )  ->  ( ( rec ( F ,  A
) `  x )  =  y  <->  <. x ,  y
>.  e.  rec ( F ,  A ) ) )
1412, 13mpan 670 . . . . . . . . . . 11  |-  ( x  e.  On  ->  (
( rec ( F ,  A ) `  x )  =  y  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) )
1511, 14syl5bb 257 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
y  =  ( rec ( F ,  A
) `  x )  <->  <.
x ,  y >.  e.  rec ( F ,  A ) ) )
1610, 15syl6 33 . . . . . . . . 9  |-  ( Lim 
B  ->  ( x  e.  B  ->  ( y  =  ( rec ( F ,  A ) `  x )  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) ) )
1716pm5.32d 639 . . . . . . . 8  |-  ( Lim 
B  ->  ( (
x  e.  B  /\  y  =  ( rec ( F ,  A ) `
 x ) )  <-> 
( x  e.  B  /\  <. x ,  y
>.  e.  rec ( F ,  A ) ) ) )
1817exbidv 1680 . . . . . . 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 258 . . . . . 6  |-  ( Lim 
B  ->  ( E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) )  <->  E. x  e.  B  y  =  ( rec ( F ,  A ) `
 x ) ) )
2019abbidv 2551 . . . . 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 2481 . . . 4  |-  ( Lim 
B  ->  ( rec ( F ,  A )
" B )  =  { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x ) } )
2221unieqd 4094 . . 3  |-  ( Lim 
B  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
2322adantl 466 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
241, 23eqtrd 2469 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 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2423   E.wrex 2710   <.cop 3876   U.cuni 4084   Ord word 4710   Oncon0 4711   Lim wlim 4712   "cima 4835    Fn wfn 5406   ` cfv 5411   reccrdg 6857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-recs 6824  df-rdg 6858
This theorem is referenced by:  rdglim2a  6881
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