Theorem setrec2 49681

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 49677 and setrec2v 49682 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 7793) 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 2891	. . . . . 6 ⊢ Ⅎ𝑎𝑥
3		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑎𝑢
4	2, 1, 3	nfbr 5142	. . . . 5 ⊢ Ⅎ𝑎 𝑥𝐹𝑢
5	4	nfeuw 2586	. . . 4 ⊢ Ⅎ𝑎∃!𝑢 𝑥𝐹𝑢
6	5	nfab 2897	. . 3 ⊢ Ⅎ𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}
7	1, 6	nfres 5936	. 2 ⊢ Ⅎ𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
8		setrec2.2	. . 3 ⊢ 𝐵 = setrecs(𝐹)
9		setrec2lem1 49679	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹‘𝑤)
10	9	sseq1i 3966	. . . . . . . . . . 11 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑧)
11	10	imbi2i 336	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12	11	imbi2i 336	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
13	12	albii 1819	. . . . . . . 8 ⊢ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
14	13	imbi1i 349	. . . . . . 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 1819	. . . . . 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 2796	. . . . 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 4873	. . . 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 49670	. . . 4 ⊢ setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
19		df-setrecs 49670	. . . 4 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
20	17, 18, 19	3eqtr4ri 2763	. . 3 ⊢ setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
21	8, 20	eqtri 2752	. 2 ⊢ 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
22		setrec2lem2 49680	. 2 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
23		setrec2.3	. . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
24		setrec2lem1 49679	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹‘𝑎)
25	24	sseq1i 3966	. . . . 5 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶 ↔ (𝐹‘𝑎) ⊆ 𝐶)
26	25	imbi2i 336	. . . 4 ⊢ ((𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ (𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
27	26	albii 1819	. . 3 ⊢ (∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
28	23, 27	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶))
29	7, 21, 22, 28	setrec2fun 49678	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540  ∃!weu 2561  {cab 2707  Ⅎwnfc 2876   ⊆ wss 3905  ∪ cuni 4861   class class class wbr 5095   ↾ cres 5625  ‘cfv 6486  setrecscsetrecs 49669

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  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-iota 6442  df-fun 6488  df-fv 6494  df-setrecs 49670

This theorem is referenced by:  setrec2v  49682  setrec2mpt  49683