Theorem setrec2fun 50503

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 50502 and setrec2v 50507 say that setrecs(𝐹) is 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 7851) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.)

Hypotheses

Ref	Expression
setrec2fun.1	⊢ Ⅎ𝑎𝐹
setrec2fun.2	⊢ 𝐵 = setrecs(𝐹)
setrec2fun.3	⊢ Fun 𝐹
setrec2fun.4	⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))

Assertion

Ref	Expression
setrec2fun	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Distinct variable group:   𝐶,𝑎

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

Proof of Theorem setrec2fun

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

Step	Hyp	Ref	Expression
1		setrec2fun.2	. . 3 ⊢ 𝐵 = setrecs(𝐹)
2		df-setrecs 50495	. . 3 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
3	1, 2	eqtri 2788	. 2 ⊢ 𝐵 = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
4		eqid 2765	. . . . . 6 ⊢ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
5		vex 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V)
7	4, 6	setrec1lem1 50498	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)))
8		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))
9		inss1 4189	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) ⊆ 𝐶
10	8, 9	sstrdi 3950	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → 𝑤 ⊆ 𝐶)
11		setrec2fun.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))
12		nfv 1947	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎 𝑤 ⊆ 𝐶
13		setrec2fun.1	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎𝐹
14		nfcv 2927	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎𝑤
15	13, 14	nffv 6895	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎(𝐹‘𝑤)
16		nfcv 2927	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎𝐶
17	15, 16	nfss 3931	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎(𝐹‘𝑤) ⊆ 𝐶
18	12, 17	nfim 1929	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎(𝑤 ⊆ 𝐶 → (𝐹‘𝑤) ⊆ 𝐶)
19		sseq1 3963	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑤 → (𝑎 ⊆ 𝐶 ↔ 𝑤 ⊆ 𝐶))
20		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (𝐹‘𝑎) = (𝐹‘𝑤))
21	20	sseq1d 3969	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑤 → ((𝐹‘𝑎) ⊆ 𝐶 ↔ (𝐹‘𝑤) ⊆ 𝐶))
22	19, 21	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑤 → ((𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶) ↔ (𝑤 ⊆ 𝐶 → (𝐹‘𝑤) ⊆ 𝐶)))
23	22	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑤 → ((𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶) → (𝑤 ⊆ 𝐶 → (𝐹‘𝑤) ⊆ 𝐶)))
24	18, 23	spimfv 2278	. . . . . . . . . . . . . . 15 ⊢ (∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶) → (𝑤 ⊆ 𝐶 → (𝐹‘𝑤) ⊆ 𝐶))
25	11, 24	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤 ⊆ 𝐶 → (𝐹‘𝑤) ⊆ 𝐶))
26	10, 25	syl5 35	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ 𝐶))
27	26	imp 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))) → (𝐹‘𝑤) ⊆ 𝐶)
28	27	3adant2 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ⊆ 𝑥 ∧ 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))) → (𝐹‘𝑤) ⊆ 𝐶)
29		velpw 4569	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝒫 𝑥 ↔ 𝑤 ⊆ 𝑥)
30		eliman0 6922	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝒫 𝑥 ∧ ¬ (𝐹‘𝑤) = ∅) → (𝐹‘𝑤) ∈ (𝐹 “ 𝒫 𝑥))
31	30	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝒫 𝑥 → (¬ (𝐹‘𝑤) = ∅ → (𝐹‘𝑤) ∈ (𝐹 “ 𝒫 𝑥)))
32	29, 31	sylbir 238	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑥 → (¬ (𝐹‘𝑤) = ∅ → (𝐹‘𝑤) ∈ (𝐹 “ 𝒫 𝑥)))
33		elssuni 4906	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤) ∈ (𝐹 “ 𝒫 𝑥) → (𝐹‘𝑤) ⊆ ∪ (𝐹 “ 𝒫 𝑥))
34	32, 33	syl6 36	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑥 → (¬ (𝐹‘𝑤) = ∅ → (𝐹‘𝑤) ⊆ ∪ (𝐹 “ 𝒫 𝑥)))
35		id 23	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤) = ∅ → (𝐹‘𝑤) = ∅)
36		0ss 4357	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ∪ (𝐹 “ 𝒫 𝑥)
37	35, 36	eqsstrdi 3982	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) = ∅ → (𝐹‘𝑤) ⊆ ∪ (𝐹 “ 𝒫 𝑥))
38	34, 37	pm2.61d2 183	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ 𝑥 → (𝐹‘𝑤) ⊆ ∪ (𝐹 “ 𝒫 𝑥))
39	38	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ⊆ 𝑥 ∧ 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))) → (𝐹‘𝑤) ⊆ ∪ (𝐹 “ 𝒫 𝑥))
40	28, 39	ssind 4193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ⊆ 𝑥 ∧ 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))
41	40	3exp 1137	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ⊆ 𝑥 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))))
42	41	alrimiv 1960	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))))
43		setrec2fun.3	. . . . . . . . . . . 12 ⊢ Fun 𝐹
44	5	pwex 5353	. . . . . . . . . . . . 13 ⊢ 𝒫 𝑥 ∈ V
45	44	funimaex 6627	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝐹 “ 𝒫 𝑥) ∈ V)
46	43, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝒫 𝑥) ∈ V
47	46	uniex 7745	. . . . . . . . . 10 ⊢ ∪ (𝐹 “ 𝒫 𝑥) ∈ V
48	47	inex2 5289	. . . . . . . . 9 ⊢ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) ∈ V
49		sseq2 3964	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))
50		sseq2 3964	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))
51	49, 50	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))))
52	51	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → ((𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑥 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))))
53	52	albidv 1953	. . . . . . . . . 10 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))))
54		sseq2 3964	. . . . . . . . . 10 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))
55	53, 54	imbi12d 347	. . . . . . . . 9 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → ((∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))) → 𝑥 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))))
56	48, 55	spcv 3566	. . . . . . . 8 ⊢ (∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝐹‘𝑤) ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))) → 𝑥 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))
57	42, 56	mpan9 516	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)) → 𝑥 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)))
58	57, 9	sstrdi 3950	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)) → 𝑥 ⊆ 𝐶)
59	58	ex 418	. . . . 5 ⊢ (𝜑 → (∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → 𝑥 ⊆ 𝐶))
60	7, 59	sylbid 243	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} → 𝑥 ⊆ 𝐶))
61	60	ralrimiv 3158	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}𝑥 ⊆ 𝐶)
62		unissb 4908	. . 3 ⊢ (∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} ⊆ 𝐶 ↔ ∀𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}𝑥 ⊆ 𝐶)
63	61, 62	sylibr 237	. 2 ⊢ (𝜑 → ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} ⊆ 𝐶)
64	3, 63	eqsstrid 3976	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2743  Ⅎwnfc 2912  ∀wral 3081  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  ∪ cuni 4874   “ cima 5666  Fun wfun 6534  ‘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

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 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-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  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-iota 6496  df-fun 6542  df-fv 6548  df-setrecs 50495

This theorem is used by:  setrec2  50506