Theorem setrec2fun 9966

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 9965 and setrec2v 9971 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 7862) 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 9958	. . 3 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
3	1, 2	eqtri 2784	. 2 ⊢ 𝐵 = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
4		eqid 2761	. . . . . 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 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V)
7	4, 6	setrec1lem1 9959	. . . . 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 4182	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) ⊆ 𝐶
10	8, 9	sstrdi 3943	. . . . . . . . . . . . . 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 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎𝑤
15	13, 14	nffv 6893	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎(𝐹‘𝑤)
16		nfcv 2923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎𝐶
17	15, 16	nfss 3924	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎(𝐹‘𝑤) ⊆ 𝐶
18	12, 17	nfim 1929	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎(𝑤 ⊆ 𝐶 → (𝐹‘𝑤) ⊆ 𝐶)
19		sseq1 3956	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑤 → (𝑎 ⊆ 𝐶 ↔ 𝑤 ⊆ 𝐶))
20		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (𝐹‘𝑎) = (𝐹‘𝑤))
21	20	sseq1d 3962	. . . . . . . . . . . . . . . . . 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 2276	. . . . . . . . . . . . . . 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 4562	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝒫 𝑥 ↔ 𝑤 ⊆ 𝑥)
30		eliman0 6920	. . . . . . . . . . . . . . . 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 4899	. . . . . . . . . . . . . 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 4350	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ ∪ (𝐹 “ 𝒫 𝑥)
37	35, 36	eqsstrdi 3975	. . . . . . . . . . . . 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 4186	. . . . . . . . . 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 5342	. . . . . . . . . . . . 13 ⊢ 𝒫 𝑥 ∈ V
45	44	funimaex 6625	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝐹 “ 𝒫 𝑥) ∈ V)
46	43, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝒫 𝑥) ∈ V
47	46	uniex 7756	. . . . . . . . . 10 ⊢ ∪ (𝐹 “ 𝒫 𝑥) ∈ V
48	47	inex2 5278	. . . . . . . . 9 ⊢ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) ∈ V
49		sseq2 3957	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥)) → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ (𝐶 ∩ ∪ (𝐹 “ 𝒫 𝑥))))
50		sseq2 3957	. . . . . . . . . . . . 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 3957	. . . . . . . . . 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 3560	. . . . . . . 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 3943	. . . . . 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 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}𝑥 ⊆ 𝐶)
62		unissb 4901	. . 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 3969	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2739  Ⅎwnfc 2908  ∀wral 3077  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867   “ cima 5654  Fun wfun 6531  ‘cfv 6537  setrecscsetrecs 9957

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fv 6545  df-setrecs 9958

This theorem is used by:  setrec2  9970