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Theorem acsfn 17602

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 6586	. . . . . . 7 ⊢ Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
2		funiunfv 7246	. . . . . . 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 4195	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎)
5	4	elpwid 4611	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ⊆ 𝑎)
6		elpwi 4609	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
7	5, 6	sylan9ssr 3996	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ⊆ 𝑋)
8		velpw 4607	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 𝑋 ↔ 𝑐 ⊆ 𝑋)
9	7, 8	sylibr 233	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
10	9	adantll 712	. . . . . . . 8 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
11		eqeq1 2736	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (𝑏 = 𝑇 ↔ 𝑐 = 𝑇))
12	11	ifbid 4551	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅))
13		eqid 2732	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
14		snex 5431	. . . . . . . . . 10 ⊢ {𝐾} ∈ V
15		0ex 5307	. . . . . . . . . 10 ⊢ ∅ ∈ V
16	14, 15	ifex 4578	. . . . . . . . 9 ⊢ if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V
17	12, 13, 16	fvmpt 6998	. . . . . . . 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 5021	. . . . . 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 2774	. . . . 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 4013	. . . 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 5048	. . . . 5 ⊢ (∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)
23		sseq1 4007	. . . . . . . . 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 4007	. . . . . . . . 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 4787	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝑋 → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎))
28	27	adantr 481	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎))
29		biimt 360	. . . . . . . . . 10 ⊢ (𝑐 = 𝑇 → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
30	29	adantl 482	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
31	28, 30	bitr3d 280	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
32		0ss 4396	. . . . . . . . . . 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 264	. . . . . . . . 9 ⊢ (¬ 𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
36	35	adantl 482	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
37	24, 26, 31, 36	ifbothda 4566	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
38	37	ralbidv 3177	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
39	38	ad3antlr 729	. . . . 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 282	. . . 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 4228	. . . . . . . 8 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
42	6	sspwd 4615	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋)
43	41, 42	sstrid 3993	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
44	43	adantl 482	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
45		ralss 4054	. . . . . 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		bi2.04 388	. . . . . . 7 ⊢ ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
48	47	ralbii 3093	. . . . . 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 4605	. . . . . . . . 9 ⊢ (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
50	49	biimparc 480	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋)
51	50	ad2antlr 725	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋)
52		eleq1 2821	. . . . . . . . 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 3513	. . . . . . 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 282	. . . . 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 776	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin)
58	57	biantrud 532	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin)))
59		elin 3964	. . . . . . . 8 ⊢ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin))
60	58, 59	bitr4di 288	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
61		vex 3478	. . . . . . . 8 ⊢ 𝑎 ∈ V
62	61	elpw2 5345	. . . . . . 7 ⊢ (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ⊆ 𝑎)
63	60, 62	bitr3di 285	. . . . . 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 304	. . . 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 305	. . 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 3439	. 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 765	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → 𝑋 ∈ 𝑉)
69		snelpwi 5443	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → {𝐾} ∈ 𝒫 𝑋)
70	69	ad2antlr 725	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋)
71		0elpw 5354	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝑋
72		ifcl 4573	. . . . . 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 481	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
75	74	fmpttd 7114	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋)
76		isacs1i 17600	. . 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 2833	1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 ↦ cmpt 5231 “ cima 5679 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 Fincfn 8938 ACScacs 17528

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-mre 17529 df-acs 17532

This theorem is referenced by: acsfn0 17603 acsfn1 17604 acsfn2 17606 acsfn1p 20414

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