Theorem acsfn 16320

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 5926	. . . . . . 7 ⊢ Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
2		funiunfv 6506	. . . . . . 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		inss1 3833	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
5	4	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎)
6	5	elpwid 4170	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ⊆ 𝑎)
7		elpwi 4168	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
8	6, 7	sylan9ssr 3617	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ⊆ 𝑋)
9		selpw 4165	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 𝑋 ↔ 𝑐 ⊆ 𝑋)
10	8, 9	sylibr 224	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
11	10	adantll 750	. . . . . . . 8 ⊢ (((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
12		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (𝑏 = 𝑇 ↔ 𝑐 = 𝑇))
13	12	ifbid 4108	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅))
14		eqid 2622	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
15		snex 4908	. . . . . . . . . 10 ⊢ {𝐾} ∈ V
16		0ex 4790	. . . . . . . . . 10 ⊢ ∅ ∈ V
17	15, 16	ifex 4156	. . . . . . . . 9 ⊢ if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V
18	13, 14, 17	fvmpt 6282	. . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑋 → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
19	11, 18	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}, ∅))
20	19	iuneq2dv 4542	. . . . . 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}, ∅))
21	3, 20	eqtr3d 2658	. . . . 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}, ∅))
22	21	sseq1d 3632	. . . 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))
23		iunss 4561	. . . . 5 ⊢ (∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)
24		sseq1 3626	. . . . . . . . 9 ⊢ ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → ({𝐾} ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
25	24	bibi1d 333	. . . . . . . 8 ⊢ ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → (({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))))
26		sseq1 3626	. . . . . . . . 9 ⊢ (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → (∅ ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
27	26	bibi1d 333	. . . . . . . 8 ⊢ (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → ((∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))))
28		snssg 4327	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝑋 → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎))
29	28	adantr 481	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎))
30		biimt 350	. . . . . . . . . 10 ⊢ (𝑐 = 𝑇 → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
31	30	adantl 482	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
32	29, 31	bitr3d 270	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
33		0ss 3972	. . . . . . . . . . 11 ⊢ ∅ ⊆ 𝑎
34	33	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑐 = 𝑇 → ∅ ⊆ 𝑎)
35		pm2.21 120	. . . . . . . . . 10 ⊢ (¬ 𝑐 = 𝑇 → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))
36	34, 35	2thd 255	. . . . . . . . 9 ⊢ (¬ 𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
37	36	adantl 482	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
38	25, 27, 32, 37	ifbothda 4123	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
39	38	ralbidv 2986	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))
40	39	ad3antlr 767	. . . . 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)))
41	23, 40	syl5bb 272	. . . 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)))
42		sspwb 4917	. . . . . . . . 9 ⊢ (𝑎 ⊆ 𝑋 ↔ 𝒫 𝑎 ⊆ 𝒫 𝑋)
43	7, 42	sylib 208	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋)
44	4, 43	syl5ss 3614	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
45	44	adantl 482	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
46		ralss 3668	. . . . . 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))))
47	45, 46	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))))
48		bi2.04 376	. . . . . . 7 ⊢ ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
49	48	ralbii 2980	. . . . . 6 ⊢ (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
50		elpwg 4166	. . . . . . . . 9 ⊢ (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
51	50	biimparc 504	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋)
52	51	ad2antlr 763	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋)
53		eleq1 2689	. . . . . . . . 9 ⊢ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
54	53	imbi1d 331	. . . . . . . 8 ⊢ (𝑐 = 𝑇 → ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
55	54	ceqsralv 3234	. . . . . . 7 ⊢ (𝑇 ∈ 𝒫 𝑋 → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)))
56	52, 55	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)))
57	49, 56	syl5bb 272	. . . . 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)))
58		vex 3203	. . . . . . . 8 ⊢ 𝑎 ∈ V
59	58	elpw2 4828	. . . . . . 7 ⊢ (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ⊆ 𝑎)
60		simplrr 801	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin)
61	60	biantrud 528	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin)))
62		elin 3796	. . . . . . . 8 ⊢ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin))
63	61, 62	syl6bbr 278	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
64	59, 63	syl5rbbr 275	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ⊆ 𝑎))
65	64	imbi1d 331	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎) ↔ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)))
66	47, 57, 65	3bitrd 294	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)))
67	22, 41, 66	3bitrrd 295	. . 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))
68	67	rabbidva 3188	. 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})
69		simpll 790	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → 𝑋 ∈ 𝑉)
70		snelpwi 4912	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → {𝐾} ∈ 𝒫 𝑋)
71	70	ad2antlr 763	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋)
72		0elpw 4834	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝑋
73		ifcl 4130	. . . . . 6 ⊢ (({𝐾} ∈ 𝒫 𝑋 ∧ ∅ ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
74	71, 72, 73	sylancl 694	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
75	74	adantr 481	. . . 4 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
76	75, 14	fmptd 6385	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋)
77		isacs1i 16318	. . 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))
78	69, 76, 77	syl2anc 693	. 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))
79	68, 78	eqeltrd 2701	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   ↦ cmpt 4729   “ cima 5117  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  Fincfn 7955  ACScacs 16245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-acs 16249

This theorem is referenced by:  acsfn0  16321  acsfn1  16322  acsfn2  16324  acsfn1p  37769