Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rdgsucg Structured version   Visualization version   GIF version

Theorem rdgsucg 7465
 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 7459 . . 3 Lim dom rec(𝐹, 𝐴)
2 limsuc 6997 . . 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 2626 . . 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 7461 . . 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 7464 . . 3 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V)
7 rdgfun 7458 . . . 4 Fun rec(𝐹, 𝐴)
8 funfn 5879 . . . 4 (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴))
97, 8mpbi 220 . . 3 rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)
10 limord 5746 . . . 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 7449 . 2 (suc 𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
133, 12sylbi 207 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 196   = wceq 1480   ∈ wcel 1992  Vcvv 3191  ∅c0 3896  ifcif 4063  ∪ cuni 4407   ↦ cmpt 4678  dom cdm 5079  ran crn 5080  Ord word 5684  Lim wlim 5686  suc csuc 5687  Fun wfun 5844   Fn wfn 5845  ‘cfv 5850  reccrdg 7451 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-wrecs 7353  df-recs 7414  df-rdg 7452 This theorem is referenced by:  rdgsuc  7466  rdgsucmptnf  7471  frsuc  7478  r1sucg  8577
 Copyright terms: Public domain W3C validator