Theorem setrec2 50629

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 50625 and setrec2v 50630 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 7853) 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 2924	. . . . . 6 ⊢ Ⅎ𝑎𝑥
3		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑎𝑢
4	2, 1, 3	nfbr 5156	. . . . 5 ⊢ Ⅎ𝑎 𝑥𝐹𝑢
5	4	nfeuw 2620	. . . 4 ⊢ Ⅎ𝑎∃!𝑢 𝑥𝐹𝑢
6	5	nfab 2930	. . 3 ⊢ Ⅎ𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}
7	1, 6	nfres 5978	. 2 ⊢ Ⅎ𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
8		setrec2.2	. . 3 ⊢ 𝐵 = setrecs(𝐹)
9		setrec2lem1 50627	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹‘𝑤)
10	9	sseq1i 3962	. . . . . . . . . . 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 1852	. . . . . . . 8 ⊢ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
14	13	imbi1i 352	. . . . . . 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 1852	. . . . . 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 2829	. . . . 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 4882	. . . 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 50618	. . . 4 ⊢ setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
19		df-setrecs 50618	. . . 4 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
20	17, 18, 19	3eqtr4ri 2796	. . 3 ⊢ setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
21	8, 20	eqtri 2785	. 2 ⊢ 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
22		setrec2lem2 50628	. 2 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
23		setrec2.3	. . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
24		setrec2lem1 50627	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹‘𝑎)
25	24	sseq1i 3962	. . . . 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 1852	. . 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 50626	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570  ∃!weu 2595  {cab 2740  Ⅎwnfc 2909   ⊆ wss 3902  ∪ cuni 4870   class class class wbr 5107   ↾ cres 5661  ‘cfv 6537  setrecscsetrecs 50617

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-setrecs 50618

This theorem is used by:  setrec2v  50630  setrec2mpt  50631