Theorem setrec2 46015

Description: This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, Theorems setrec1 46011 and setrec2v 46016 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7610) to the other class.

(Contributed by Emmett Weisz, 2-Sep-2021.)

Hypotheses

Ref	Expression
setrec2.1	⊢ Ⅎ𝑎𝐹
setrec2.2	⊢ 𝐵 = setrecs(𝐹)
setrec2.3	⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))

Assertion

Ref	Expression
setrec2	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Distinct variable group:   𝐶,𝑎

Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)   𝐹(𝑎)

Proof of Theorem setrec2

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

Step	Hyp	Ref	Expression
1		setrec2.1	. . 3 ⊢ Ⅎ𝑎𝐹
2		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑎𝑥
3		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑎𝑢
4	2, 1, 3	nfbr 5086	. . . . 5 ⊢ Ⅎ𝑎 𝑥𝐹𝑢
5	4	nfeuw 2592	. . . 4 ⊢ Ⅎ𝑎∃!𝑢 𝑥𝐹𝑢
6	5	nfab 2903	. . 3 ⊢ Ⅎ𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}
7	1, 6	nfres 5838	. 2 ⊢ Ⅎ𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
8		setrec2.2	. . 3 ⊢ 𝐵 = setrecs(𝐹)
9		setrec2lem1 46013	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹‘𝑤)
10	9	sseq1i 3915	. . . . . . . . . . 11 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑧)
11	10	imbi2i 339	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12	11	imbi2i 339	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
13	12	albii 1827	. . . . . . . 8 ⊢ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
14	13	imbi1i 353	. . . . . . 7 ⊢ ((∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧))
15	14	albii 1827	. . . . . 6 ⊢ (∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧))
16	15	abbii 2801	. . . . 5 ⊢ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	16	unieqi 4818	. . . 4 ⊢ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
18		df-setrecs 46004	. . . 4 ⊢ setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
19		df-setrecs 46004	. . . 4 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
20	17, 18, 19	3eqtr4ri 2770	. . 3 ⊢ setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
21	8, 20	eqtri 2759	. 2 ⊢ 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
22		setrec2lem2 46014	. 2 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
23		setrec2.3	. . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
24		setrec2lem1 46013	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹‘𝑎)
25	24	sseq1i 3915	. . . . 5 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶 ↔ (𝐹‘𝑎) ⊆ 𝐶)
26	25	imbi2i 339	. . . 4 ⊢ ((𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ (𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
27	26	albii 1827	. . 3 ⊢ (∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
28	23, 27	sylibr 237	. 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶))
29	7, 21, 22, 28	setrec2fun 46012	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4  ∀wal 1541   = wceq 1543  ∃!weu 2567  {cab 2714  Ⅎwnfc 2877   ⊆ wss 3853  ∪ cuni 4805   class class class wbr 5039   ↾ cres 5538  ‘cfv 6358  setrecscsetrecs 46003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fv 6366  df-setrecs 46004

This theorem is referenced by:  setrec2v  46016