Theorem rdgseg 8298

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 8286	. . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2	1	reseq1i 5904	. 2 ⊢ (rec(𝐹, 𝐴) ↾ 𝐵) = (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) ↾ 𝐵)
3		rdglem1 8291	. . . 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 8262	. . 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 5831	. . 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 2856	. 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 2842	1 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2714  ∀wral 3062  ∃wrex 3071  Vcvv 3441  ∅c0 4266  ifcif 4469  ∪ cuni 4848   ↦ cmpt 5168  dom cdm 5605  ran crn 5606   ↾ cres 5607  Oncon0 6286  Lim wlim 6287   Fn wfn 6458  ‘cfv 6463  recscrecs 8246  reccrdg 8285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fo 6469  df-fv 6471  df-ov 7316  df-2nd 7875  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286

This theorem is referenced by:  rdgsucg  8299  rdglimg  8301