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Theorem setrec1 42763
Description: This is the first of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is closed under 𝐹. This effectively sets the actual value of setrecs(𝐹) as a lower bound for setrecs(𝐹), as it implies that any set generated by successive applications of 𝐹 is a member of 𝐵. This theorem "gets off the ground" because we can start by letting 𝐴 = ∅, and the hypotheses of the theorem will hold trivially.

Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon).

Proof summary: Assume that 𝐴𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 42761, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 42761.) Therefore, by setrec1lem4 42762, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4491, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 42756 for a detailed description of how the setrecs definition works.

(Contributed by Emmett Weisz, 9-Oct-2020.)

Hypotheses
Ref Expression
setrec1.b 𝐵 = setrecs(𝐹)
setrec1.v (𝜑𝐴 ∈ V)
setrec1.a (𝜑𝐴𝐵)
Assertion
Ref Expression
setrec1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)

Proof of Theorem setrec1
Dummy variables 𝑎 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . 4 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2 setrec1.v . . . 4 (𝜑𝐴 ∈ V)
3 setrec1.a . . . . . . . . 9 (𝜑𝐴𝐵)
43sseld 3635 . . . . . . . 8 (𝜑 → (𝑎𝐴𝑎𝐵))
54imp 444 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐵)
6 setrec1.b . . . . . . . 8 𝐵 = setrecs(𝐹)
7 df-setrecs 42756 . . . . . . . 8 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
86, 7eqtri 2673 . . . . . . 7 𝐵 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
95, 8syl6eleq 2740 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 eluni 4471 . . . . . 6 (𝑎 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} ↔ ∃𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
119, 10sylib 208 . . . . 5 ((𝜑𝑎𝐴) → ∃𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
1211ralrimiva 2995 . . . 4 (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
131, 2, 12setrec1lem3 42761 . . 3 (𝜑 → ∃𝑥(𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
14 nfv 1883 . . . . . . 7 𝑧𝜑
15 nfv 1883 . . . . . . . 8 𝑧 𝐴𝑥
16 nfaba1 2799 . . . . . . . . 9 𝑧{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1716nfel2 2810 . . . . . . . 8 𝑧 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1815, 17nfan 1868 . . . . . . 7 𝑧(𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
1914, 18nfan 1868 . . . . . 6 𝑧(𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
202adantr 480 . . . . . 6 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → 𝐴 ∈ V)
21 simprl 809 . . . . . 6 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → 𝐴𝑥)
22 simprr 811 . . . . . 6 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
2319, 1, 20, 21, 22setrec1lem4 42762 . . . . 5 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → (𝑥 ∪ (𝐹𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
24 ssun2 3810 . . . . 5 (𝐹𝐴) ⊆ (𝑥 ∪ (𝐹𝐴))
2523, 24jctil 559 . . . 4 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → ((𝐹𝐴) ⊆ (𝑥 ∪ (𝐹𝐴)) ∧ (𝑥 ∪ (𝐹𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
26 ssuni 4491 . . . 4 (((𝐹𝐴) ⊆ (𝑥 ∪ (𝐹𝐴)) ∧ (𝑥 ∪ (𝐹𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}) → (𝐹𝐴) ⊆ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
2725, 26syl 17 . . 3 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → (𝐹𝐴) ⊆ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
2813, 27exlimddv 1903 . 2 (𝜑 → (𝐹𝐴) ⊆ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
2928, 8syl6sseqr 3685 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  Vcvv 3231  cun 3605  wss 3607   cuni 4468  cfv 5926  setrecscsetrecs 42755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666  df-setrecs 42756
This theorem is referenced by:  elsetrecslem  42770  elsetrecs  42771  setrecsss  42772  setrecsres  42773  vsetrec  42774  onsetrec  42779
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