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Theorem rdgsucg 7107
Description: The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdgsucg  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )

Proof of Theorem rdgsucg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgdmlim 7101 . . 3  |-  Lim  dom  rec ( F ,  A
)
2 limsuc 6683 . . 3  |-  ( Lim 
dom  rec ( F ,  A )  ->  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) ) )
31, 2ax-mp 5 . 2  |-  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) )
4 eqid 2457 . . 3  |-  ( 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
) ) ) ) )
5 rdgvalg 7103 . . 3  |-  ( 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 ) ) )
6 rdgseg 7106 . . 3  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  y )  e.  _V )
7 rdgfun 7100 . . . 4  |-  Fun  rec ( F ,  A )
8 funfn 5623 . . . 4  |-  ( Fun 
rec ( F ,  A )  <->  rec ( F ,  A )  Fn  dom  rec ( F ,  A ) )
97, 8mpbi 208 . . 3  |-  rec ( F ,  A )  Fn  dom  rec ( F ,  A )
10 limord 4946 . . . 4  |-  ( Lim 
dom  rec ( F ,  A )  ->  Ord  dom 
rec ( F ,  A ) )
111, 10ax-mp 5 . . 3  |-  Ord  dom  rec ( F ,  A
)
124, 5, 6, 9, 11tz7.44-2 7091 . 2  |-  ( suc 
B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
133, 12sylbi 195 1  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ifcif 3944   U.cuni 4251    |-> cmpt 4515   Ord word 4886   Lim wlim 4888   suc csuc 4889   dom cdm 5008   ran crn 5009   Fun wfun 5588    Fn wfn 5589   ` cfv 5594   reccrdg 7093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-recs 7060  df-rdg 7094
This theorem is referenced by:  rdgsuc  7108  rdgsucmptnf  7113  frsuc  7120  r1sucg  8204
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