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Theorem mreacs 17599

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 6889	. . 3 ⊢ (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
2		pweq 4616	. . . 4 ⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
3	2	fveq2d 6893	. . 3 ⊢ (𝑥 = 𝑋 → (Moore‘𝒫 𝑥) = (Moore‘𝒫 𝑋))
4	1, 3	eleq12d 2828	. 2 ⊢ (𝑥 = 𝑋 → ((ACS‘𝑥) ∈ (Moore‘𝒫 𝑥) ↔ (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)))
5		acsmre 17593	. . . . . . 7 ⊢ (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ (Moore‘𝑥))
6		mresspw 17533	. . . . . . . 8 ⊢ (𝑎 ∈ (Moore‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ (ACS‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
8	5, 7	elpwd 4608	. . . . . 6 ⊢ (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ 𝒫 𝒫 𝑥)
9	8	ssriv 3986	. . . . 5 ⊢ (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥
10	9	a1i 11	. . . 4 ⊢ (⊤ → (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥)
11		vex 3479	. . . . . . . 8 ⊢ 𝑥 ∈ V
12		mremre 17545	. . . . . . . 8 ⊢ (𝑥 ∈ V → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
13	11, 12	mp1i 13	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
14	5	ssriv 3986	. . . . . . . 8 ⊢ (ACS‘𝑥) ⊆ (Moore‘𝑥)
15		sstr 3990	. . . . . . . 8 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ (ACS‘𝑥) ⊆ (Moore‘𝑥)) → 𝑎 ⊆ (Moore‘𝑥))
16	14, 15	mpan2 690	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → 𝑎 ⊆ (Moore‘𝑥))
17		mrerintcl 17538	. . . . . . 7 ⊢ (((Moore‘𝑥) ∈ (Moore‘𝒫 𝑥) ∧ 𝑎 ⊆ (Moore‘𝑥)) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (Moore‘𝑥))
18	13, 16, 17	syl2anc 585	. . . . . 6 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (Moore‘𝑥))
19		ssel2 3977	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑑 ∈ (ACS‘𝑥))
20	19	acsmred 17597	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑑 ∈ (Moore‘𝑥))
21		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (mrCls‘𝑑) = (mrCls‘𝑑)
22	20, 21	mrcssvd 17564	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
23	22	ralrimiva 3147	. . . . . . . . . . . . 13 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
24	23	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
25		iunss 5048	. . . . . . . . . . . 12 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥 ↔ ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
26	24, 25	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
27	11	elpw2 5345	. . . . . . . . . . 11 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥 ↔ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
28	26, 27	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥)
29	28	fmpttd 7112	. . . . . . . . 9 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥)
30		fssxp 6743	. . . . . . . . 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))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥))
32		vpwex 5375	. . . . . . . . 9 ⊢ 𝒫 𝑥 ∈ V
33	32, 32	xpex 7737	. . . . . . . 8 ⊢ (𝒫 𝑥 × 𝒫 𝑥) ∈ V
34		ssexg 5323	. . . . . . . 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)
35	31, 33, 34	sylancl 587	. . . . . . 7 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) ∈ V)
36	19	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑑 ∈ (ACS‘𝑥))
37		elpwi 4609	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝒫 𝑥 → 𝑏 ⊆ 𝑥)
38	37	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑 ∈ 𝑎) → 𝑏 ⊆ 𝑥)
39	21	acsfiel2 17596	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ACS‘𝑥) ∧ 𝑏 ⊆ 𝑥) → (𝑏 ∈ 𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
40	36, 38, 39	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑 ∈ 𝑎) → (𝑏 ∈ 𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
41	40	ralbidva 3176	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
42		iunss 5048	. . . . . . . . . . . . 13 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
43	42	ralbii 3094	. . . . . . . . . . . 12 ⊢ (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
44		ralcom 3287	. . . . . . . . . . . 12 ⊢ (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
45	43, 44	bitri 275	. . . . . . . . . . 11 ⊢ (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
46	41, 45	bitr4di 289	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
47		elrint2 4996	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝒫 𝑥 → (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑))
48	47	adantl 483	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∀𝑑 ∈ 𝑎 𝑏 ∈ 𝑑))
49		funmpt 6584	. . . . . . . . . . . . 13 ⊢ Fun (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))
50		funiunfv 7244	. . . . . . . . . . . . 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)))
51	49, 50	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))
52	51	sseq1i 4010	. . . . . . . . . . 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)
53		iunss 5048	. . . . . . . . . . . 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)
54		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐))
55		fveq2 6889	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑒 → ((mrCls‘𝑑)‘𝑐) = ((mrCls‘𝑑)‘𝑒))
56	55	iuneq2d 5026	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐) = ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒))
57		inss1 4228	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑏
58	37	sspwd 4615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝒫 𝑥 → 𝒫 𝑏 ⊆ 𝒫 𝑥)
59	58	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → 𝒫 𝑏 ⊆ 𝒫 𝑥)
60	57, 59	sstrid 3993	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑥)
61	60	sselda 3982	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑒 ∈ 𝒫 𝑥)
62	20, 21	mrcssvd 17564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑 ∈ 𝑎) → ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
63	62	ralrimiva 3147	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
64	63	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
65		iunss 5048	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥 ↔ ∀𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
66	64, 65	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
67		ssexg 5323	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥 ∧ 𝑥 ∈ V) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
68	66, 11, 67	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
69	54, 56, 61, 68	fvmptd3 7019	. . . . . . . . . . . . . 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))
70	69	sseq1d 4013	. . . . . . . . . . . . 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))
71	70	ralbidva 3176	. . . . . . . . . . . 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))
72	53, 71	bitrid 283	. . . . . . . . . . 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))
73	52, 72	bitr3id 285	. . . . . . . . . 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))
74	46, 48, 73	3bitr4d 311	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
75	74	ralrimiva 3147	. . . . . . . 8 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
76	29, 75	jca 513	. . . . . . 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)))
77		feq1 6696	. . . . . . . 8 ⊢ (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))
78		imaeq1 6053	. . . . . . . . . . . 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)))
79	78	unieqd 4922	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) → ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ∪ ((𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
80	79	sseq1d 4013	. . . . . . . . . 10 ⊢ (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))
81	80	bibi2d 343	. . . . . . . . 9 ⊢ (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)))
82	81	ralbidv 3178	. . . . . . . 8 ⊢ (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)))
83	77, 82	anbi12d 632	. . . . . . 7 ⊢ (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))))
84	35, 76, 83	spcedv 3589	. . . . . 6 ⊢ (𝑎 ⊆ (ACS‘𝑥) → ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 ∩ ∩ 𝑎) ↔ ∪ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
85		isacs 17592	. . . . . 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))))
86	18, 84, 85	sylanbrc 584	. . . . 5 ⊢ (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (ACS‘𝑥))
87	86	adantl 483	. . . 4 ⊢ ((⊤ ∧ 𝑎 ⊆ (ACS‘𝑥)) → (𝒫 𝑥 ∩ ∩ 𝑎) ∈ (ACS‘𝑥))
88	10, 87	ismred2 17544	. . 3 ⊢ (⊤ → (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥))
89	88	mptru 1549	. 2 ⊢ (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥)
90	4, 89	vtoclg 3557	1 ⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 ∩ cint 4950 ∪ ciun 4997 ↦ cmpt 5231 × cxp 5674 “ cima 5679 Fun wfun 6535 ⟶wf 6537 ‘cfv 6541 Fincfn 8936 Moorecmre 17523 mrClscmrc 17524 ACScacs 17526

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 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 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-mre 17527 df-mrc 17528 df-acs 17530

This theorem is referenced by: acsfn1 17602 acsfn1c 17603 acsfn2 17604 submacs 18705 subgacs 19036 nsgacs 19037 acsfn1p 20408 subrgacs 20409 sdrgacs 20410 lssacs 20571

W3C validator