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Theorem rdglimg 7385
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 2609 . 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 7379 . 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 7382 . 2 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V)
4 rdgfun 7376 . . 3 Fun rec(𝐹, 𝐴)
5 funfn 5818 . . 3 (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴))
64, 5mpbi 218 . 2 rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)
7 rdgdmlim 7377 . . 3 Lim dom rec(𝐹, 𝐴)
8 limord 5686 . . 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 7368 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 382   = wceq 1474  wcel 1976  Vcvv 3172  c0 3873  ifcif 4035   cuni 4366  cmpt 4637  dom cdm 5027  ran crn 5028  cima 5030  Ord word 5624  Lim wlim 5626  Fun wfun 5783   Fn wfn 5784  cfv 5789  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by:  rdglim  7386  r1limg  8494
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