Theorem ismrcd1 42659

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17554), isotone (satisfies mrcss 17553), and idempotent (satisfies mrcidm 17556) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 42660 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
ismrcd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
ismrcd.f	⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
ismrcd.e	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
ismrcd.m	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
ismrcd.i	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))

Assertion

Ref	Expression
ismrcd1	⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦

Proof of Theorem ismrcd1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4196	. . . 4 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
2		dmss 5856	. . . 4 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . 3 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
4		ismrcd.f	. . 3 ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
5	3, 4	fssdm 6689	. 2 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
6		ssid 3966	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
7		ismrcd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8		elpwg 4562	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
10	6, 9	mpbiri 258	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
11	4, 10	ffvelcdmd 7039	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝒫 𝐵)
12	11	elpwid 4568	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ⊆ 𝐵)
13		velpw 4564	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
14		ismrcd.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
15	13, 14	sylan2b 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
16	15	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
19	17, 18	sseq12d 3977	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝐵 ⊆ (𝐹‘𝐵)))
20	19	rspcva 3583	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → 𝐵 ⊆ (𝐹‘𝐵))
21	10, 16, 20	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐹‘𝐵))
22	12, 21	eqssd 3961	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐵)
23	4	ffnd 6671	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
24		fnelfp 7131	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
25	23, 10, 24	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
26	22, 25	mpbird 257	. 2 ⊢ (𝜑 → 𝐵 ∈ dom (𝐹 ∩ I ))
27		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ dom (𝐹 ∩ I ))
28	5	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
29	27, 28	sstrd 3954	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ 𝒫 𝐵)
30		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ≠ ∅)
31		intssuni2 4933	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝒫 𝐵 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
32	29, 30, 31	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
33		unipw 5405	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐵 = 𝐵
34	32, 33	sseqtrdi 3984	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ 𝐵)
35		intex 5294	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ ∅ ↔ ∩ 𝑧 ∈ V)
36		elpwg 4562	. . . . . . . . . . . 12 ⊢ (∩ 𝑧 ∈ V → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
37	35, 36	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
38	37	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
39	34, 38	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ 𝒫 𝐵)
40	39	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ∈ 𝒫 𝐵)
41		ismrcd.m	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
42	41	3expib 1122	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
43	42	alrimiv 1927	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
44	43	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
45	44	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
46	29	sselda 3943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝒫 𝐵)
47	46	elpwid 4568	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ⊆ 𝐵)
48		intss1 4923	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑧 → ∩ 𝑧 ⊆ 𝑥)
49	48	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ⊆ 𝑥)
50	47, 49	jca 511	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥))
51		sseq1 3969	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝑦 ⊆ 𝑥 ↔ ∩ 𝑧 ⊆ 𝑥))
52	51	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥)))
53		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝐹‘𝑦) = (𝐹‘∩ 𝑧))
54	53	sseq1d 3975	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥)))
55	52, 54	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑧 → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
56	55	spcgv 3559	. . . . . . . 8 ⊢ (∩ 𝑧 ∈ 𝒫 𝐵 → (∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
57	40, 45, 50, 56	syl3c 66	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))
58	27	sselda 3943	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ dom (𝐹 ∩ I ))
59	23	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝐹 Fn 𝒫 𝐵)
60	59	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝐹 Fn 𝒫 𝐵)
61		fnelfp 7131	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
62	60, 46, 61	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
63	58, 62	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘𝑥) = 𝑥)
64	57, 63	sseqtrd 3980	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ 𝑥)
65	64	ralrimiva 3125	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
66		ssint 4924	. . . . 5 ⊢ ((𝐹‘∩ 𝑧) ⊆ ∩ 𝑧 ↔ ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
67	65, 66	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) ⊆ ∩ 𝑧)
68	16	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
69		id 22	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → 𝑥 = ∩ 𝑧)
70		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → (𝐹‘𝑥) = (𝐹‘∩ 𝑧))
71	69, 70	sseq12d 3977	. . . . . 6 ⊢ (𝑥 = ∩ 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧)))
72	71	rspcva 3583	. . . . 5 ⊢ ((∩ 𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
73	39, 68, 72	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
74	67, 73	eqssd 3961	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) = ∩ 𝑧)
75		fnelfp 7131	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∩ 𝑧 ∈ 𝒫 𝐵) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
76	59, 39, 75	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
77	74, 76	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ dom (𝐹 ∩ I ))
78	5, 26, 77	ismred 17539	1 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  ∪ cuni 4867  ∩ cint 4906   I cid 5525  dom cdm 5631   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  Moorecmre 17519

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-mre 17523

This theorem is referenced by:  ismrcd2  42660  istopclsd  42661  ismrc  42662