Theorem rdgseg 7860

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 7848	. . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2	1	reseq1i 5688	. 2 ⊢ (rec(𝐹, 𝐴) ↾ 𝐵) = (recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) ↾ 𝐵)
3		rdglem1 7853	. . . 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 7824	. . 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 5619	. . 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 2877	. 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	syl5eqel 2863	1 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2751  ∀wral 3081  ∃wrex 3082  Vcvv 3408  ∅c0 4172  ifcif 4344  ∪ cuni 4708   ↦ cmpt 5004  dom cdm 5403  ran crn 5404   ↾ cres 5405  Oncon0 6026  Lim wlim 6027   Fn wfn 6180  ‘cfv 6185  recscrecs 7809  reccrdg 7847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193  df-wrecs 7748  df-recs 7810  df-rdg 7848

This theorem is referenced by:  rdgsucg  7861  rdglimg  7863