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Theorem setrec1 44696
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 44694, 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 44694.) Therefore, by setrec1lem4 44695, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4859, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 44689 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 2826 . . . 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 3970 . . . . . . . 8 (𝜑 → (𝑎𝐴𝑎𝐵))
54imp 407 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐵)
6 setrec1.b . . . . . . . 8 𝐵 = setrecs(𝐹)
7 df-setrecs 44689 . . . . . . . 8 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
86, 7eqtri 2849 . . . . . . 7 𝐵 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
95, 8syl6eleq 2928 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 eluni 4840 . . . . . 6 (𝑎 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} ↔ ∃𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
119, 10sylib 219 . . . . 5 ((𝜑𝑎𝐴) → ∃𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
1211ralrimiva 3187 . . . 4 (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
131, 2, 12setrec1lem3 44694 . . 3 (𝜑 → ∃𝑥(𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}))
14 nfv 1908 . . . . . . 7 𝑧𝜑
15 nfv 1908 . . . . . . . 8 𝑧 𝐴𝑥
16 nfaba1 2991 . . . . . . . . 9 𝑧{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1716nfel2 3001 . . . . . . . 8 𝑧 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1815, 17nfan 1893 . . . . . . 7 𝑧(𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
1914, 18nfan 1893 . . . . . 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 767 . . . . . 6 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → 𝐴𝑥)
22 simprr 769 . . . . . 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 44695 . . . . 5 ((𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) → (𝑥 ∪ (𝐹𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
24 ssun2 4153 . . . . 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 4859 . . . 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 1929 . 2 (𝜑 → (𝐹𝐴) ⊆ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
2928, 8sseqtrrdi 4022 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107  {cab 2804  Vcvv 3500  cun 3938  wss 3940   cuni 4837  cfv 6354  setrecscsetrecs 44688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-reg 9050  ax-inf2 9098
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-r1 9187  df-rank 9188  df-setrecs 44689
This theorem is referenced by:  elsetrecslem  44703  elsetrecs  44704  setrecsss  44705  setrecsres  44706  vsetrec  44707  onsetrec  44712
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