Theorem mreacs 16142

Description: Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
mreacs	⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

Proof of Theorem mreacs

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6103	. . 3 ⊢ (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
2		pweq 4111	. . . 4 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
3	2	fveq2d 6107	. . 3 ⊢ (𝑥 = 𝑋 → (Moore‘𝒫 𝑥) = (Moore‘𝒫 𝑋))
4	1, 3	eleq12d 2682	. 2 ⊢ (𝑥 = 𝑋 → ((ACS‘𝑥) ∈ (Moore‘𝒫 𝑥) ↔ (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)))
5		acsmre 16136	. . . . . . . 8 ⊢ (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ (Moore‘𝑥))
6		mresspw 16075	. . . . . . . 8 ⊢ (𝑎 ∈ (Moore‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ (ACS‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
8		selpw 4115	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝒫 𝑥 ↔ 𝑎 ⊆ 𝒫 𝑥)
9	7, 8	sylibr 223	. . . . . 6 ⊢ (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ 𝒫 𝒫 𝑥)
10	9	ssriv 3572	. . . . 5 ⊢ (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥
11	10	a1i 11	. . . 4 ⊢ (⊤ → (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥)
12		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
13		mremre 16087	. . . . . . . 8 ⊢ (𝑥 ∈ V → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
14	12, 13	mp1i 13	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
15	5	ssriv 3572	. . . . . . . 8 ⊢ (ACS‘𝑥) ⊆ (Moore‘𝑥)
16		sstr 3576	. . . . . . . 8 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ (ACS‘𝑥) ⊆ (Moore‘𝑥)) → 𝑎 ⊆ (Moore‘𝑥))
17	15, 16	mpan2 703	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → 𝑎 ⊆ (Moore‘𝑥))
18		mrerintcl 16080	. . . . . . 7 ⊢ (((Moore‘𝑥) ∈ (Moore‘𝒫 𝑥) ∧ 𝑎 ⊆ (Moore‘𝑥)) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (Moore‘𝑥))
19	14, 17, 18	syl2anc 691	. . . . . 6 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (Moore‘𝑥))
20		ssel2 3563	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑑 ∈ (ACS‘𝑥))
21	20	acsmred 16140	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑑 ∈ (Moore‘𝑥))
22		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (mrCls‘𝑑) = (mrCls‘𝑑)
23	21, 22	mrcssvd 16106	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
24	23	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
26		iunss 4497	. . . . . . . . . . . 12 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥 ↔ ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
27	25, 26	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
28	12	elpw2 4755	. . . . . . . . . . 11 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥 ↔ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
29	27, 28	sylibr 223	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥)
30		eqid 2610	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))
31	29, 30	fmptd 6292	. . . . . . . . 9 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥)
32		fssxp 5973	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥))
33	31, 32	syl 17	. . . . . . . 8 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥))
34		vpwex 4775	. . . . . . . . 9 ⊢ 𝒫 𝑥 ∈ V
35	34, 34	xpex 6860	. . . . . . . 8 ⊢ (𝒫 𝑥 × 𝒫 𝑥) ∈ V
36		ssexg 4732	. . . . . . . 8 ⊢ (((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥) ∧ (𝒫 𝑥 × 𝒫 𝑥) ∈ V) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ∈ V)
37	33, 35, 36	sylancl 693	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ∈ V)
38	20	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑑 ∈ (ACS‘𝑥))
39		elpwi 4117	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝒫 𝑥 → 𝑏 ⊆ 𝑥)
40	39	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑏 ⊆ 𝑥)
41	22	acsfiel2 16139	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ACS‘𝑥) ∧ 𝑏 ⊆ 𝑥) → (𝑏 ∈ 𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
42	38, 40, 41	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑 ∈ 𝑎) → (𝑏 ∈ 𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
43	42	ralbidva 2968	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
44		iunss 4497	. . . . . . . . . . . . 13 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
45	44	ralbii 2963	. . . . . . . . . . . 12 ⊢ (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
46		ralcom 3079	. . . . . . . . . . . 12 ⊢ (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
47	45, 46	bitri 263	. . . . . . . . . . 11 ⊢ (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
48	43, 47	syl6bbr 277	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
49		elrint2 4454	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝒫 𝑥 → (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑))
50	49	adantl 481	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑))
51		funmpt 5840	. . . . . . . . . . . . 13 ⊢ Fun (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))
52		funiunfv 6410	. . . . . . . . . . . . 13 ⊢ (Fun (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → ∪ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
53	51, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ ∪ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin))
54	53	sseq1i 3592	. . . . . . . . . . 11 ⊢ (∪ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)
55		iunss 4497	. . . . . . . . . . . 12 ⊢ (∪ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏)
56		inss1 3795	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑏
57		sspwb 4844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ⊆ 𝑥 ↔ 𝒫 𝑏 ⊆ 𝒫 𝑥)
58	39, 57	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝒫 𝑥 → 𝒫 𝑏 ⊆ 𝒫 𝑥)
59	58	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → 𝒫 𝑏 ⊆ 𝒫 𝑥)
60	56, 59	syl5ss 3579	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑥)
61	60	sselda 3568	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑒 ∈ 𝒫 𝑥)
62	21, 22	mrcssvd 16106	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
63	62	ralrimiva 2949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
64	63	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
65		iunss 4497	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥 ↔ ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
66	64, 65	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
67		ssexg 4732	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥 ∧ 𝑥 ∈ V) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
68	66, 12, 67	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
69		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑒 → ((mrCls‘𝑑)‘𝑐) = ((mrCls‘𝑑)‘𝑒))
70	69	iuneq2d 4483	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑒 → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) = ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒))
71	70, 30	fvmptg 6189	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 ∈ 𝒫 𝑥 ∧ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V) → ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒))
72	61, 68, 71	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒))
73	72	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → (((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
74	73	ralbidva 2968	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
75	55, 74	syl5bb 271	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∪ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
76	54, 75	syl5bbr 273	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
77	48, 50, 76	3bitr4d 299	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
78	77	ralrimiva 2949	. . . . . . . 8 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
79	31, 78	jca 553	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
80		feq1 5939	. . . . . . . . 9 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓:𝒫 𝑥⟶𝒫 𝑥 ↔ (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥))
81		imaeq1 5380	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
82	81	unieqd 4382	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
83	82	sseq1d 3595	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → (∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏 ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
84	83	bibi2d 331	. . . . . . . . . 10 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → ((𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏) ↔ (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
85	84	ralbidv 2969	. . . . . . . . 9 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → (∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏) ↔ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
86	80, 85	anbi12d 743	. . . . . . . 8 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → ((𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)) ↔ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))))
87	86	spcegv 3267	. . . . . . 7 ⊢ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ∈ V → (((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)) → ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))))
88	37, 79, 87	sylc 63	. . . . . 6 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
89		isacs 16135	. . . . . 6 ⊢ ((𝒫 𝑥 ∩ ∩ 𝑎) ∈ (ACS‘𝑥) ↔ ((𝒫 𝑥 ∩ ∩ 𝑎) ∈ (Moore‘𝑥) ∧ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))))
90	19, 88, 89	sylanbrc 695	. . . . 5 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (ACS‘𝑥))
91	90	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑎 ⊆ (ACS‘𝑥)) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (ACS‘𝑥))
92	11, 91	ismred2 16086	. . 3 ⊢ (⊤ → (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥))
93	92	trud 1484	. 2 ⊢ (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥)
94	4, 93	vtoclg 3239	1 ⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070  df-acs 16072

This theorem is referenced by:  acsfn1  16145  acsfn1c  16146  acsfn2  16147  submacs  17188  subgacs  17452  nsgacs  17453  lssacs  18788  acsfn1p  36788  subrgacs  36789  sdrgacs  36790