Theorem setrec1 50502

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 50500, 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 50500.) Therefore, by setrec1lem4 50501, (𝐹‘𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4900, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 50495 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 2765	. . . 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 3937	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝐵))
5	4	imp 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐵)
6		setrec1.b	. . . . . . . 8 ⊢ 𝐵 = setrecs(𝐹)
7		df-setrecs 50495	. . . . . . . 8 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
8	6, 7	eqtri 2788	. . . . . . 7 ⊢ 𝐵 = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
9	5, 8	eleqtrdi 2875	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
10		eluni 4877	. . . . . 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 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
12	11	ralrimiva 3159	. . . 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 50500	. . 3 ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
14		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑧𝜑
15		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑧 𝐴 ⊆ 𝑥
16		nfaba1 2935	. . . . . . . . 9 ⊢ Ⅎ𝑧{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	16	nfel2 2945	. . . . . . . 8 ⊢ Ⅎ𝑧 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
18	15, 17	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑧(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
19	14, 18	nfan 1932	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}))
20	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → 𝐴 ∈ V)
21		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) → 𝐴 ⊆ 𝑥)
22		simprr 785	. . . . . 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 50501	. . . . 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 529	. . . 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 4900	. . . 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 18	. . 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 1968	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  ∪ cuni 4874  ‘cfv 6540  setrecscsetrecs 50494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-reg 9557  ax-inf2 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-r1 9739  df-rank 9740  df-scott 9861  df-setrecs 50495

This theorem is used by:  elsetrecslem  50510  elsetrecs  50511  setrecsss  50512  setrecsres  50513  vsetrec  50514  onsetrec  50519