Theorem setrec1 49680

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 𝐴 ⊆ 𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 49678, 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 49678.) Therefore, by setrec1lem4 49679, (𝐹‘𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4886, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 49673 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	⊢ (𝜑 → 𝐴 ∈ V)
setrec1.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
setrec1	⊢ (𝜑 → (𝐹‘𝐴) ⊆ 𝐵)

Proof of Theorem setrec1

Dummy variables 𝑎 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
2		setrec1.v	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
3		setrec1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	sseld 3936	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝐵))
5	4	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐵)
6		setrec1.b	. . . . . . . 8 ⊢ 𝐵 = setrecs(𝐹)
7		df-setrecs 49673	. . . . . . . 8 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
8	6, 7	eqtri 2752	. . . . . . 7 ⊢ 𝐵 = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
9	5, 8	eleqtrdi 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
10		eluni 4864	. . . . . 6 ⊢ (𝑎 ∈ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
11	9, 10	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
12	11	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
13	1, 2, 12	setrec1lem3 49678	. . 3 ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
14		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑧𝜑
15		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑧 𝐴 ⊆ 𝑥
16		nfaba1 2899	. . . . . . . . 9 ⊢ Ⅎ𝑧{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	16	nfel2 2910	. . . . . . . 8 ⊢ Ⅎ𝑧 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
18	15, 17	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑧(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
19	14, 18	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
20	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → 𝐴 ∈ V)
21		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → 𝐴 ⊆ 𝑥)
22		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
23	19, 1, 20, 21, 22	setrec1lem4 49679	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → (𝑥 ∪ (𝐹‘𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
24		ssun2 4132	. . . . 5 ⊢ (𝐹‘𝐴) ⊆ (𝑥 ∪ (𝐹‘𝐴))
25	23, 24	jctil 519	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → ((𝐹‘𝐴) ⊆ (𝑥 ∪ (𝐹‘𝐴)) ∧ (𝑥 ∪ (𝐹‘𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
26		ssuni 4886	. . . 4 ⊢ (((𝐹‘𝐴) ⊆ (𝑥 ∪ (𝐹‘𝐴)) ∧ (𝑥 ∪ (𝐹‘𝐴)) ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}) → (𝐹‘𝐴) ⊆ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
27	25, 26	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → (𝐹‘𝐴) ⊆ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
28	13, 27	exlimddv 1935	. 2 ⊢ (𝜑 → (𝐹‘𝐴) ⊆ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
29	28, 8	sseqtrrdi 3979	1 ⊢ (𝜑 → (𝐹‘𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3438   ∪ cun 3903   ⊆ wss 3905  ∪ cuni 4861  ‘cfv 6486  setrecscsetrecs 49672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-reg 9503  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679  df-rank 9680  df-setrecs 49673

This theorem is referenced by:  elsetrecslem  49688  elsetrecs  49689  setrecsss  49690  setrecsres  49691  vsetrec  49692  onsetrec  49697