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Theorem rdglim2 7100
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 7094 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. ( rec ( F ,  A ) " B ) )
2 dfima3 5330 . . . . 5  |-  ( rec ( F ,  A
) " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) ) }
3 df-rex 2799 . . . . . . 7  |-  ( E. x  e.  B  y  =  ( rec ( F ,  A ) `  x )  <->  E. x
( x  e.  B  /\  y  =  ( rec ( F ,  A
) `  x )
) )
4 limord 4927 . . . . . . . . . . 11  |-  ( Lim 
B  ->  Ord  B )
5 ordelord 4890 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  x  e.  B )  ->  Ord  x )
65ex 434 . . . . . . . . . . . 12  |-  ( Ord 
B  ->  ( x  e.  B  ->  Ord  x
) )
7 vex 3098 . . . . . . . . . . . . 13  |-  x  e. 
_V
87elon 4877 . . . . . . . . . . . 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 2452 . . . . . . . . . . 11  |-  ( y  =  ( rec ( F ,  A ) `  x )  <->  ( rec ( F ,  A ) `
 x )  =  y )
12 rdgfnon 7086 . . . . . . . . . . . 12  |-  rec ( F ,  A )  Fn  On
13 fnopfvb 5899 . . . . . . . . . . . 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 1701 . . . . . . 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 2579 . . . . 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 2496 . . . 4  |-  ( Lim 
B  ->  ( rec ( F ,  A )
" B )  =  { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x ) } )
2221unieqd 4244 . . 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 2484 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 1383   E.wex 1599    e. wcel 1804   {cab 2428   E.wrex 2794   <.cop 4020   U.cuni 4234   Ord word 4867   Oncon0 4868   Lim wlim 4869   "cima 4992    Fn wfn 5573   ` cfv 5578   reccrdg 7077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-recs 7044  df-rdg 7078
This theorem is referenced by:  rdglim2a  7101
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