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Theorem rdgsucg 7281
Description: The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdgsucg (𝐵 ∈ 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 7275 . . 3 Lim dom rec(𝐹, 𝐴)
2 limsuc 6816 . . 3 (Lim dom rec(𝐹, 𝐴) → (𝐵 ∈ dom rec(𝐹, 𝐴) ↔ suc 𝐵 ∈ dom rec(𝐹, 𝐴)))
31, 2ax-mp 5 . 2 (𝐵 ∈ dom rec(𝐹, 𝐴) ↔ suc 𝐵 ∈ dom rec(𝐹, 𝐴))
4 eqid 2514 . . 3 (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥))))) = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))
5 rdgvalg 7277 . . 3 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝑦) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ 𝑦)))
6 rdgseg 7280 . . 3 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V)
7 rdgfun 7274 . . . 4 Fun rec(𝐹, 𝐴)
8 funfn 5718 . . . 4 (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴))
97, 8mpbi 218 . . 3 rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)
10 limord 5589 . . . 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 7265 . 2 (suc 𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
133, 12sylbi 205 1 (𝐵 ∈ 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 194   = wceq 1474  wcel 1938  Vcvv 3077  c0 3777  ifcif 3939   cuni 4270  cmpt 4541  dom cdm 4932  ran crn 4933  Ord word 5529  Lim wlim 5531  suc csuc 5532  Fun wfun 5683   Fn wfn 5684  cfv 5689  reccrdg 7267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-wrecs 7168  df-recs 7230  df-rdg 7268
This theorem is referenced by:  rdgsuc  7282  rdgsucmptnf  7287  frsuc  7294  r1sucg  8390
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