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

Theorem rdglimg 6675
Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdglimg  |-  ( ( B  e.  dom  rec ( F ,  A )  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. ( rec ( F ,  A ) " B
) )

Proof of Theorem rdglimg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . 2  |-  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `  U. dom  x ) ) ) ) )  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `
 U. dom  x
) ) ) ) )
2 rdgvalg 6669 . 2  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  y )  =  ( ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `
 U. dom  x
) ) ) ) ) `  ( rec ( F ,  A
)  |`  y ) ) )
3 rdgseg 6672 . 2  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  y )  e.  _V )
4 rdgfun 6666 . . 3  |-  Fun  rec ( F ,  A )
5 funfn 5474 . . 3  |-  ( Fun 
rec ( F ,  A )  <->  rec ( F ,  A )  Fn  dom  rec ( F ,  A ) )
64, 5mpbi 200 . 2  |-  rec ( F ,  A )  Fn  dom  rec ( F ,  A )
7 rdgdmlim 6667 . . 3  |-  Lim  dom  rec ( F ,  A
)
8 limord 4632 . . 3  |-  ( Lim 
dom  rec ( F ,  A )  ->  Ord  dom 
rec ( F ,  A ) )
97, 8ax-mp 8 . 2  |-  Ord  dom  rec ( F ,  A
)
101, 2, 3, 6, 9tz7.44-3 6658 1  |-  ( ( B  e.  dom  rec ( F ,  A )  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. ( rec ( F ,  A ) " B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   ifcif 3731   U.cuni 4007    e. cmpt 4258   Ord word 4572   Lim wlim 4574   dom cdm 4870   ran crn 4871   "cima 4873   Fun wfun 5440    Fn wfn 5441   ` cfv 5446   reccrdg 6659
This theorem is referenced by:  rdglim  6676  r1limg  7689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
  Copyright terms: Public domain W3C validator