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Theorem rdglim2 7053
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 7047 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A
) `  B )  =  U. ( rec ( F ,  A ) " B ) )
2 dfima3 5279 . . . . 5  |-  ( rec ( F ,  A
) " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  rec ( F ,  A ) ) }
3 df-rex 2757 . . . . . . 7  |-  ( E. x  e.  B  y  =  ( rec ( F ,  A ) `  x )  <->  E. x
( x  e.  B  /\  y  =  ( rec ( F ,  A
) `  x )
) )
4 limord 4878 . . . . . . . . . . 11  |-  ( Lim 
B  ->  Ord  B )
5 ordelord 4841 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  x  e.  B )  ->  Ord  x )
65ex 432 . . . . . . . . . . . 12  |-  ( Ord 
B  ->  ( x  e.  B  ->  Ord  x
) )
7 vex 3059 . . . . . . . . . . . . 13  |-  x  e. 
_V
87elon 4828 . . . . . . . . . . . 12  |-  ( x  e.  On  <->  Ord  x )
96, 8syl6ibr 227 . . . . . . . . . . 11  |-  ( Ord 
B  ->  ( x  e.  B  ->  x  e.  On ) )
104, 9syl 17 . . . . . . . . . 10  |-  ( Lim 
B  ->  ( x  e.  B  ->  x  e.  On ) )
11 eqcom 2409 . . . . . . . . . . 11  |-  ( y  =  ( rec ( F ,  A ) `  x )  <->  ( rec ( F ,  A ) `
 x )  =  y )
12 rdgfnon 7039 . . . . . . . . . . . 12  |-  rec ( F ,  A )  Fn  On
13 fnopfvb 5844 . . . . . . . . . . . 12  |-  ( ( rec ( F ,  A )  Fn  On  /\  x  e.  On )  ->  ( ( rec ( F ,  A
) `  x )  =  y  <->  <. x ,  y
>.  e.  rec ( F ,  A ) ) )
1412, 13mpan 668 . . . . . . . . . . 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 31 . . . . . . . . 9  |-  ( Lim 
B  ->  ( x  e.  B  ->  ( y  =  ( rec ( F ,  A ) `  x )  <->  <. x ,  y >.  e.  rec ( F ,  A ) ) ) )
1716pm5.32d 637 . . . . . . . 8  |-  ( Lim 
B  ->  ( (
x  e.  B  /\  y  =  ( rec ( F ,  A ) `
 x ) )  <-> 
( x  e.  B  /\  <. x ,  y
>.  e.  rec ( F ,  A ) ) ) )
1817exbidv 1733 . . . . . . 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 2536 . . . . 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 2453 . . . 4  |-  ( Lim 
B  ->  ( rec ( F ,  A )
" B )  =  { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x ) } )
2221unieqd 4198 . . 3  |-  ( Lim 
B  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
2322adantl 464 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  U. ( rec ( F ,  A
) " B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x
) } )
241, 23eqtrd 2441 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 367    = wceq 1403   E.wex 1631    e. wcel 1840   {cab 2385   E.wrex 2752   <.cop 3975   U.cuni 4188   Ord word 4818   Oncon0 4819   Lim wlim 4820   "cima 4943    Fn wfn 5518   ` cfv 5523   reccrdg 7030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-recs 6997  df-rdg 7031
This theorem is referenced by:  rdglim2a  7054
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