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Theorem rdgsucg 7406
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 7400 . . 3 Lim dom rec(𝐹, 𝐴)
2 limsuc 6941 . . 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 2610 . . 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 7402 . . 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 7405 . . 3 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V)
7 rdgfun 7399 . . . 4 Fun rec(𝐹, 𝐴)
8 funfn 5833 . . . 4 (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴))
97, 8mpbi 219 . . 3 rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)
10 limord 5701 . . . 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 7390 . 2 (suc 𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
133, 12sylbi 206 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 195   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  ifcif 4036   cuni 4372  cmpt 4643  dom cdm 5038  ran crn 5039  Ord word 5639  Lim wlim 5641  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  cfv 5804  reccrdg 7392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393
This theorem is referenced by:  rdgsuc  7407  rdgsucmptnf  7412  frsuc  7419  r1sucg  8515
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