Theorem isacs1i 16677

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 3914	. . . 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		inss1 4059	. . . . . 6 ⊢ (𝑋 ∩ ∩ 𝑡) ⊆ 𝑋
4		elpw2g 5051	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋 ↔ (𝑋 ∩ ∩ 𝑡) ⊆ 𝑋))
5	3, 4	mpbiri 250	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋)
6	5	ad2antrr 717	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋)
7		imassrn 5722	. . . . . . . . 9 ⊢ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ran 𝐹
8		frn 6288	. . . . . . . . . 10 ⊢ (𝐹:𝒫 𝑋⟶𝒫 𝑋 → ran 𝐹 ⊆ 𝒫 𝑋)
9	8	adantl 475	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ran 𝐹 ⊆ 𝒫 𝑋)
10	7, 9	syl5ss 3838	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
11	10	unissd 4686	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∪ 𝒫 𝑋)
12		unipw 5141	. . . . . . 7 ⊢ ∪ 𝒫 𝑋 = 𝑋
13	11, 12	syl6sseq 3876	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑋)
14	13	adantr 474	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑋)
15		inss2 4060	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∩ ∩ 𝑡) ⊆ ∩ 𝑡
16		intss1 4714	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝑡 → ∩ 𝑡 ⊆ 𝑎)
17	15, 16	syl5ss 3838	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑡 → (𝑋 ∩ ∩ 𝑡) ⊆ 𝑎)
18	17	adantl 475	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → (𝑋 ∩ ∩ 𝑡) ⊆ 𝑎)
19		sspwb 5140	. . . . . . . . . . . 12 ⊢ ((𝑋 ∩ ∩ 𝑡) ⊆ 𝑎 ↔ 𝒫 (𝑋 ∩ ∩ 𝑡) ⊆ 𝒫 𝑎)
20	18, 19	sylib 210	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → 𝒫 (𝑋 ∩ ∩ 𝑡) ⊆ 𝒫 𝑎)
21	20	ssrind 4066	. . . . . . . . . 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))
22		imass2 5746	. . . . . . . . . 10 ⊢ ((𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin) → (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
23	21, 22	syl 17	. . . . . . . . 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)))
24	23	unissd 4686	. . . . . . . 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)))
25		ssel2 3822	. . . . . . . . . 10 ⊢ ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎 ∈ 𝑡) → 𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
26		pweq 4383	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → 𝒫 𝑠 = 𝒫 𝑎)
27	26	ineq1d 4042	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑎 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑎 ∩ Fin))
28	27	imaeq2d 5711	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑎 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
29	28	unieqd 4670	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑎 → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
30		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑎 → 𝑠 = 𝑎)
31	29, 30	sseq12d 3859	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑎 → (∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 ↔ ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
32	31	elrab 3585	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
33	32	simprbi 492	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} → ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
34	25, 33	syl 17	. . . . . . . . 9 ⊢ ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎 ∈ 𝑡) → ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
35	34	adantll 705	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎 ∈ 𝑡) → ∪ (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
36	24, 35	sstrd 3837	. . . . . . 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)
37	36	ralrimiva 3175	. . . . . 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)
38		ssint 4715	. . . . . 6 ⊢ (∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∩ 𝑡 ↔ ∀𝑎 ∈ 𝑡 ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ 𝑎)
39	37, 38	sylibr 226	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ ∩ 𝑡)
40	14, 39	ssind 4063	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝑋 ∩ ∩ 𝑡))
41		pweq 4383	. . . . . . . . 9 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → 𝒫 𝑠 = 𝒫 (𝑋 ∩ ∩ 𝑡))
42	41	ineq1d 4042	. . . . . . . 8 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → (𝒫 𝑠 ∩ Fin) = (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin))
43	42	imaeq2d 5711	. . . . . . 7 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)))
44	43	unieqd 4670	. . . . . 6 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)))
45		id 22	. . . . . 6 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → 𝑠 = (𝑋 ∩ ∩ 𝑡))
46	44, 45	sseq12d 3859	. . . . 5 ⊢ (𝑠 = (𝑋 ∩ ∩ 𝑡) → (∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 ↔ ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝑋 ∩ ∩ 𝑡)))
47	46	elrab 3585	. . . 4 ⊢ ((𝑋 ∩ ∩ 𝑡) ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ((𝑋 ∩ ∩ 𝑡) ∈ 𝒫 𝑋 ∧ ∪ (𝐹 “ (𝒫 (𝑋 ∩ ∩ 𝑡) ∩ Fin)) ⊆ (𝑋 ∩ ∩ 𝑡)))
48	6, 40, 47	sylanbrc 578	. . 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})
49	2, 48	ismred2 16623	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋))
50		fssxp 6301	. . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝒫 𝑋 → 𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋))
51		pwexg 5080	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
52	51, 51	xpexd 7226	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 × 𝒫 𝑋) ∈ V)
53		ssexg 5031	. . . 4 ⊢ ((𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋) ∧ (𝒫 𝑋 × 𝒫 𝑋) ∈ V) → 𝐹 ∈ V)
54	50, 52, 53	syl2anr 590	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹 ∈ V)
55		simpr 479	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
56		pweq 4383	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → 𝒫 𝑠 = 𝒫 𝑡)
57	56	ineq1d 4042	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑡 ∩ Fin))
58	57	imaeq2d 5711	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
59	58	unieqd 4670	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)))
60		id 22	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
61	59, 60	sseq12d 3859	. . . . . 6 ⊢ (𝑠 = 𝑡 → (∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
62	61	elrab3 3587	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
63	62	rgen 3131	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)
64	55, 63	jctir 516	. . 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)))
65		feq1 6263	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋 ↔ 𝐹:𝒫 𝑋⟶𝒫 𝑋))
66		imaeq1 5706	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
67	66	unieqd 4670	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)))
68	67	sseq1d 3857	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
69	68	bibi2d 334	. . . . . 6 ⊢ (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)))
70	69	ralbidv 3195	. . . . 5 ⊢ (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)))
71	65, 70	anbi12d 624	. . . 4 ⊢ (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))))
72	71	spcegv 3511	. . 3 ⊢ (𝐹 ∈ V → ((𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
73	54, 64, 72	sylc 65	. 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)))
74		isacs 16671	. 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))))
75	49, 73, 74	sylanbrc 578	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∀wral 3117  {crab 3121  Vcvv 3414   ∩ cin 3797   ⊆ wss 3798  𝒫 cpw 4380  ∪ cuni 4660  ∩ cint 4699   × cxp 5344  ran crn 5347   “ cima 5349  ⟶wf 6123  ‘cfv 6127  Fincfn 8228  Moorecmre 16602  ACScacs 16605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-mre 16606  df-acs 16609

This theorem is referenced by:  acsfn  16679