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Theorem rdglimg 7566
 Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdglimg ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))

Proof of Theorem rdglimg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . 2 (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)))))
2 rdgvalg 7560 . 2 (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)))
3 rdgseg 7563 . 2 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V)
4 rdgfun 7557 . . 3 Fun rec(𝐹, 𝐴)
5 funfn 5956 . . 3 (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴))
64, 5mpbi 220 . 2 rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)
7 rdgdmlim 7558 . . 3 Lim dom rec(𝐹, 𝐴)
8 limord 5822 . . 3 (Lim dom rec(𝐹, 𝐴) → Ord dom rec(𝐹, 𝐴))
97, 8ax-mp 5 . 2 Ord dom rec(𝐹, 𝐴)
101, 2, 3, 6, 9tz7.44-3 7549 1 ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  ifcif 4119  ∪ cuni 4468   ↦ cmpt 4762  dom cdm 5143  ran crn 5144   “ cima 5146  Ord word 5760  Lim wlim 5762  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  reccrdg 7550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-wrecs 7452  df-recs 7513  df-rdg 7551 This theorem is referenced by:  rdglim  7567  r1limg  8672
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