Theorem isacs1i 17738

Description: A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
isacs1i	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))

Distinct variable groups:   𝐹,𝑠   𝑋,𝑠

Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem isacs1i

Dummy variables 𝑎 𝑡 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4037	. . . 4 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ⊆ 𝒫 𝑋
2	1	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ⊆ 𝒫 𝑋)
3		pweq 4581	. . . . . . . 8 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → 𝒫 𝑠 = 𝒫 (𝑋 ∩ ∩ 𝑡))
4	3	ineq1d 4175	. . . . . . 7 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → (𝒫 𝑠 ∩ Fin) = (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin))
5	4	imaeq2d 6067	. . . . . 6 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)))
6	5	unieqd 4890	. . . . 5 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)))
7		id 23	. . . . 5 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → 𝑠 = (𝑋 ∩ ∩ 𝑡))
8	6, 7	sseq12d 3973	. . . 4 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → (∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 ↔ ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝑋 ∩ ∩ 𝑡)))
9		inss1 4192	. . . . . 6 ⊢ (𝑋 ∩ ∩ 𝑡) ⊆ 𝑋
10		elpw2g 5309	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋 ↔ (𝑋 ∩ ∩ 𝑡) ⊆ 𝑋))
11	9, 10	mpbiri 261	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋)
12	11	ad2antrr 739	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋)
13		imassrn 6078	. . . . . . . . 9 ⊢ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ran 𝐹
14		frn 6720	. . . . . . . . . 10 ⊢ (𝐹:𝒫 𝑋⟶𝒫 𝑋 → ran 𝐹 ⊆ 𝒫 𝑋)
15	14	adantl 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ran 𝐹 ⊆ 𝒫 𝑋)
16	13, 15	sstrid 3951	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
17	16	unissd 4887	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∪ 𝒫 𝑋)
18		unipw 5436	. . . . . . 7 ⊢ ∪ 𝒫 𝑋 = 𝑋
19	17, 18	sseqtrdi 3980	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑋)
20	19	adantr 486	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑋)
21		inss2 4193	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∩ ∩ 𝑡) ⊆ ∩ 𝑡
22		intss1 4933	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝑡 → ∩ 𝑡 ⊆ 𝑎)
23	21, 22	sstrid 3951	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑡 → (𝑋 ∩ ∩ 𝑡) ⊆ 𝑎)
24	23	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → (𝑋 ∩ ∩ 𝑡) ⊆ 𝑎)
25	24	sspwd 4580	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → 𝒫 (𝑋 ∩ ∩ 𝑡) ⊆ 𝒫 𝑎)
26	25	ssrind 4199	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin))
27		imass2 6109	. . . . . . . . . 10 ⊢ ((𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin) → (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
28	26, 27	syl 18	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
29	28	unissd 4887	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
30		ssel2 3935	. . . . . . . . . 10 ⊢ ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎 ∈ 𝑡) → 𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
31		pweq 4581	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → 𝒫 𝑠 = 𝒫 𝑎)
32	31	ineq1d 4175	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑎 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑎 ∩ Fin))
33	32	imaeq2d 6067	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑎 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
34	33	unieqd 4890	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑎 → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
35		id 23	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑎 → 𝑠 = 𝑎)
36	34, 35	sseq12d 3973	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑎 → (∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 ↔ ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
37	36	elrab 3653	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
38	37	simprbi 503	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} → ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
39	30, 38	syl 18	. . . . . . . . 9 ⊢ ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎 ∈ 𝑡) → ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
40	39	adantll 727	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
41	29, 40	sstrd 3950	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑎)
42	41	ralrimiva 3160	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∀𝑎 ∈ 𝑡 ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑎)
43		ssint 4934	. . . . . 6 ⊢ (∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∩ 𝑡 ↔ ∀𝑎 ∈ 𝑡 ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑎)
44	42, 43	sylibr 237	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∩ 𝑡)
45	20, 44	ssind 4196	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝑋 ∩ ∩ 𝑡))
46	8, 12, 45	elrabd 3655	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 ∩ ∩ 𝑡) ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
47	2, 46	ismred2 17680	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋))
48		fssxp 6740	. . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝒫 𝑋 → 𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋))
49		pwexg 5354	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
50	49, 49	xpexd 7759	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 × 𝒫 𝑋) ∈ V)
51		ssexg 5295	. . . 4 ⊢ ((𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋) ∧ (𝒫 𝑋 × 𝒫 𝑋) ∈ V) → 𝐹 ∈ V)
52	48, 50, 51	syl2anr 609	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹 ∈ V)
53		simpr 490	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
54		pweq 4581	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → 𝒫 𝑠 = 𝒫 𝑡)
55	54	ineq1d 4175	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑡 ∩ Fin))
56	55	imaeq2d 6067	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
57	56	unieqd 4890	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)))
58		id 23	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
59	57, 58	sseq12d 3973	. . . . . 6 ⊢ (𝑠 = 𝑡 → (∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
60	59	elrab3 3654	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
61	60	rgen 3084	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)
62	53, 61	jctir 530	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
63		feq1 6690	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋 ↔ 𝐹:𝒫 𝑋⟶𝒫 𝑋))
64		imaeq1 6062	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
65	64	unieqd 4890	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)))
66	65	sseq1d 3971	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
67	66	bibi2d 345	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
68	67	ralbidv 3191	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
69	63, 68	anbi12d 644	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
70	52, 62, 69	spcedv 3560	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
71		isacs 17732	. 2 ⊢ ({𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋) ↔ ({𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
72	47, 70, 71	sylanbrc 595	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  {crab 3419  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877  ∩ cint 4917   × cxp 5664  ran crn 5667   “ cima 5669  ⟶wf 6539  ‘cfv 6543  Fincfn 8952  Moorecmre 17659  ACScacs 17662

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 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17663  df-acs 17666

This theorem is used by:  acsfn  17740