Theorem acsfn 17620

Description: Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
acsfn	⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋))

Distinct variable groups:   𝐾,𝑎   𝑇,𝑎   𝑉,𝑎   𝑋,𝑎

Proof of Theorem acsfn

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6554	. . . . . . 7 ⊢ Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
2		funiunfv 7222	. . . . . . 7 ⊢ (Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) → ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)))
3	1, 2	mp1i 13	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)))
4		elinel1 4164	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎)
5	4	elpwid 4572	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ⊆ 𝑎)
6		elpwi 4570	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
7	5, 6	sylan9ssr 3961	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ⊆ 𝑋)
8		velpw 4568	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 𝑋 ↔ 𝑐 ⊆ 𝑋)
9	7, 8	sylibr 234	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
10	9	adantll 714	. . . . . . . 8 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
11		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (𝑏 = 𝑇 ↔ 𝑐 = 𝑇))
12	11	ifbid 4512	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅))
13		eqid 2729	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
14		snex 5391	. . . . . . . . . 10 ⊢ {𝐾} ∈ V
15		0ex 5262	. . . . . . . . . 10 ⊢ ∅ ∈ V
16	14, 15	ifex 4539	. . . . . . . . 9 ⊢ if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V
17	12, 13, 16	fvmpt 6968	. . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑋 → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
18	10, 17	syl 17	. . . . . . 7 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
19	18	iuneq2dv 4980	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅))
20	3, 19	eqtr3d 2766	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) = ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅))
21	20	sseq1d 3978	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎 ↔ ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
22		iunss 5009	. . . . 5 ⊢ (∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)
23		sseq1 3972	. . . . . . . . 9 ⊢ ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → ({𝐾} ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
24	23	bibi1d 343	. . . . . . . 8 ⊢ ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → (({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))))
25		sseq1 3972	. . . . . . . . 9 ⊢ (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → (∅ ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
26	25	bibi1d 343	. . . . . . . 8 ⊢ (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → ((∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))))
27		snssg 4747	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝑋 → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎))
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎))
29		biimt 360	. . . . . . . . . 10 ⊢ (𝑐 = 𝑇 → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
30	29	adantl 481	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
31	28, 30	bitr3d 281	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
32		0ss 4363	. . . . . . . . . . 11 ⊢ ∅ ⊆ 𝑎
33	32	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑐 = 𝑇 → ∅ ⊆ 𝑎)
34		pm2.21 123	. . . . . . . . . 10 ⊢ (¬ 𝑐 = 𝑇 → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))
35	33, 34	2thd 265	. . . . . . . . 9 ⊢ (¬ 𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
36	35	adantl 481	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
37	24, 26, 31, 36	ifbothda 4527	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
38	37	ralbidv 3156	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
39	38	ad3antlr 731	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
40	22, 39	bitrid 283	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
41		inss1 4200	. . . . . . . 8 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
42	6	sspwd 4576	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋)
43	41, 42	sstrid 3958	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
44	43	adantl 481	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
45		ralss 4021	. . . . . 6 ⊢ ((𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))))
46	44, 45	syl 17	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))))
47		bi2.04 387	. . . . . . 7 ⊢ ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
48	47	ralbii 3075	. . . . . 6 ⊢ (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
49		elpwg 4566	. . . . . . . . 9 ⊢ (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
50	49	biimparc 479	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋)
51	50	ad2antlr 727	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋)
52		eleq1 2816	. . . . . . . . 9 ⊢ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
53	52	imbi1d 341	. . . . . . . 8 ⊢ (𝑐 = 𝑇 → ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
54	53	ceqsralv 3488	. . . . . . 7 ⊢ (𝑇 ∈ 𝒫 𝑋 → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
55	51, 54	syl 17	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
56	48, 55	bitrid 283	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
57		simplrr 777	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin)
58	57	biantrud 531	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin)))
59		elin 3930	. . . . . . . 8 ⊢ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin))
60	58, 59	bitr4di 289	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
61		vex 3451	. . . . . . . 8 ⊢ 𝑎 ∈ V
62	61	elpw2 5289	. . . . . . 7 ⊢ (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ⊆ 𝑎)
63	60, 62	bitr3di 286	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ⊆ 𝑎))
64	63	imbi1d 341	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎) ↔ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)))
65	46, 56, 64	3bitrd 305	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)))
66	21, 40, 65	3bitrrd 306	. . 3 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎) ↔ ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
67	66	rabbidva 3412	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎})
68		simpll 766	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → 𝑋 ∈ 𝑉)
69		snelpwi 5403	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → {𝐾} ∈ 𝒫 𝑋)
70	69	ad2antlr 727	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋)
71		0elpw 5311	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝑋
72		ifcl 4534	. . . . . 6 ⊢ (({𝐾} ∈ 𝒫 𝑋 ∧ ∅ ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
73	70, 71, 72	sylancl 586	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
74	73	adantr 480	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
75	74	fmpttd 7087	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋)
76		isacs1i 17618	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋))
77	68, 75, 76	syl2anc 584	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋))
78	67, 77	eqeltrd 2828	1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ifcif 4488  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5188   “ cima 5641  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  Fincfn 8918  ACScacs 17546

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-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-mre 17547  df-acs 17550

This theorem is referenced by:  acsfn0  17621  acsfn1  17622  acsfn2  17624  acsfn1p  20708