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Theorem rdgseg 7688
Description: The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
rdgseg (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)

Proof of Theorem rdgseg
Dummy variables 𝑥 𝑦 𝑓 𝑔 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rdg 7676 . . 3 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
21reseq1i 5547 . 2 (rec(𝐹, 𝐴) ↾ 𝐵) = (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) ↾ 𝐵)
3 rdglem1 7681 . . . 4 {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑣𝑦 (𝑤𝑣) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑤𝑣)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)))}
43tfrlem9a 7652 . . 3 (𝐵 ∈ dom recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) → (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) ↾ 𝐵) ∈ V)
51dmeqi 5480 . . 3 dom rec(𝐹, 𝐴) = dom recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
64, 5eleq2s 2857 . 2 (𝐵 ∈ dom rec(𝐹, 𝐴) → (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) ↾ 𝐵) ∈ V)
72, 6syl5eqel 2843 1 (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  c0 4058  ifcif 4230   cuni 4588  cmpt 4881  dom cdm 5266  ran crn 5267  cres 5268  Oncon0 5884  Lim wlim 5885   Fn wfn 6044  cfv 6049  recscrecs 7637  reccrdg 7675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-wrecs 7577  df-recs 7638  df-rdg 7676
This theorem is referenced by:  rdgsucg  7689  rdglimg  7691
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