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Theorem setrec1 46397
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 46395, 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 46395.) Therefore, by setrec1lem4 46396, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4866, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 46390 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 2738 . . . 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 3920 . . . . . . . 8 (𝜑 → (𝑎𝐴𝑎𝐵))
54imp 407 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐵)
6 setrec1.b . . . . . . . 8 𝐵 = setrecs(𝐹)
7 df-setrecs 46390 . . . . . . . 8 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
86, 7eqtri 2766 . . . . . . 7 𝐵 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
95, 8eleqtrdi 2849 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 eluni 4842 . . . . . 6 (𝑎 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} ↔ ∃𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
119, 10sylib 217 . . . . 5 ((𝜑𝑎𝐴) → ∃𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
1211ralrimiva 3103 . . . 4 (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
131, 2, 12setrec1lem3 46395 . . 3 (𝜑 → ∃𝑥(𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
14 nfv 1917 . . . . . . 7 𝑧𝜑
15 nfv 1917 . . . . . . . 8 𝑧 𝐴𝑥
16 nfaba1 2915 . . . . . . . . 9 𝑧{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1716nfel2 2925 . . . . . . . 8 𝑧 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1815, 17nfan 1902 . . . . . . 7 𝑧(𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
1914, 18nfan 1902 . . . . . 6 𝑧(𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
202adantr 481 . . . . . 6 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → 𝐴 ∈ V)
21 simprl 768 . . . . . 6 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → 𝐴𝑥)
22 simprr 770 . . . . . 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 46396 . . . . 5 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → (𝑥 ∪ (𝐹𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
24 ssun2 4107 . . . . 5 (𝐹𝐴) ⊆ (𝑥 ∪ (𝐹𝐴))
2523, 24jctil 520 . . . 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 4866 . . . 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 1938 . 2 (𝜑 → (𝐹𝐴) ⊆ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
2928, 8sseqtrrdi 3972 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537   = wceq 1539  wex 1782  wcel 2106  {cab 2715  Vcvv 3432  cun 3885  wss 3887   cuni 4839  cfv 6433  setrecscsetrecs 46389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523  df-setrecs 46390
This theorem is referenced by:  elsetrecslem  46404  elsetrecs  46405  setrecsss  46406  setrecsres  46407  vsetrec  46408  onsetrec  46413
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