Theorem rdgseg 8478

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.

Step	Hyp	Ref	Expression
1		df-rdg 8466	. . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2	1	reseq1i 6005	. 2 ⊢ (rec(𝐹, 𝐴) ↾ 𝐵) = (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) ↾ 𝐵)
3		rdglem1 8471	. . . 4 ⊢ {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑣 ∈ 𝑦 (𝑤‘𝑣) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑤 ↾ 𝑣)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)))}
4	3	tfrlem9a 8442	. . 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)
5	1	dmeqi 5929	. . 3 ⊢ dom rec(𝐹, 𝐴) = dom recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
6	4, 5	eleq2s 2862	. 2 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) ↾ 𝐵) ∈ V)
7	2, 6	eqeltrid 2848	1 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488  ∅c0 4352  ifcif 4548  ∪ cuni 4931   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   ↾ cres 5702  Oncon0 6395  Lim wlim 6396   Fn wfn 6568  ‘cfv 6573  recscrecs 8426  reccrdg 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  rdgsucg  8479  rdglimg  8481