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Theorem rdgsucg 6644
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 6638 . . 3  |-  Lim  dom  rec ( F ,  A
)
2 limsuc 4792 . . 3  |-  ( Lim 
dom  rec ( F ,  A )  ->  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) ) )
31, 2ax-mp 8 . 2  |-  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) )
4 eqid 2408 . . 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 6640 . . 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 6643 . . 3  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  y )  e.  _V )
7 rdgfun 6637 . . . 4  |-  Fun  rec ( F ,  A )
8 funfn 5445 . . . 4  |-  ( Fun 
rec ( F ,  A )  <->  rec ( F ,  A )  Fn  dom  rec ( F ,  A ) )
97, 8mpbi 200 . . 3  |-  rec ( F ,  A )  Fn  dom  rec ( F ,  A )
10 limord 4604 . . . 4  |-  ( Lim 
dom  rec ( F ,  A )  ->  Ord  dom 
rec ( F ,  A ) )
111, 10ax-mp 8 . . 3  |-  Ord  dom  rec ( F ,  A
)
124, 5, 6, 9, 11tz7.44-2 6628 . 2  |-  ( suc 
B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
133, 12sylbi 188 1  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   _Vcvv 2920   (/)c0 3592   ifcif 3703   U.cuni 3979    e. cmpt 4230   Ord word 4544   Lim wlim 4546   suc csuc 4547   dom cdm 4841   ran crn 4842   Fun wfun 5411    Fn wfn 5412   ` cfv 5417   reccrdg 6630
This theorem is referenced by:  rdgsuc  6645  rdgsucmptnf  6650  frsuc  6657  r1sucg  7655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631
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