Theorem ismrcd1 43147

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17574), isotone (satisfies mrcss 17573), and idempotent (satisfies mrcidm 17576) 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 43148 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 4165	. . . 4 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
2		dmss 5844	. . . 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 6674	. 2 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
6		ssid 3937	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
7		ismrcd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8		elpwg 4532	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
10	6, 9	mpbiri 259	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
11	4, 10	ffvelcdmd 7026	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝒫 𝐵)
12	11	elpwid 4538	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ⊆ 𝐵)
13		velpw 4534	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
14		ismrcd.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
15	13, 14	sylan2b 600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
16	15	ralrimiva 3131	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		fveq2 6827	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
19	17, 18	sseq12d 3948	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝐵 ⊆ (𝐹‘𝐵)))
20	19	rspcva 3558	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → 𝐵 ⊆ (𝐹‘𝐵))
21	10, 16, 20	syl2anc 590	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐹‘𝐵))
22	12, 21	eqssd 3932	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐵)
23	4	ffnd 6656	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
24		fnelfp 7119	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
25	23, 10, 24	syl2anc 590	. . 3 ⊢ (𝜑 → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
26	22, 25	mpbird 258	. 2 ⊢ (𝜑 → 𝐵 ∈ dom (𝐹 ∩ I ))
27		simp2 1143	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ dom (𝐹 ∩ I ))
28	5	3ad2ant1 1139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
29	27, 28	sstrd 3925	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ 𝒫 𝐵)
30		simp3 1144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ≠ ∅)
31		intssuni2 4903	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝒫 𝐵 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
32	29, 30, 31	syl2anc 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
33		unipw 5389	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐵 = 𝐵
34	32, 33	sseqtrdi 3955	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ 𝐵)
35		intex 5272	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ ∅ ↔ ∩ 𝑧 ∈ V)
36		elpwg 4532	. . . . . . . . . . . 12 ⊢ (∩ 𝑧 ∈ V → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
37	35, 36	sylbi 218	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
38	37	3ad2ant3 1141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
39	34, 38	mpbird 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ 𝒫 𝐵)
40	39	adantr 481	. . . . . . . 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 1128	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
43	42	alrimiv 1934	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
44	43	3ad2ant1 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
45	44	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
46	29	sselda 3915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝒫 𝐵)
47	46	elpwid 4538	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ⊆ 𝐵)
48		intss1 4893	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑧 → ∩ 𝑧 ⊆ 𝑥)
49	48	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ⊆ 𝑥)
50	47, 49	jca 516	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥))
51		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝑦 ⊆ 𝑥 ↔ ∩ 𝑧 ⊆ 𝑥))
52	51	anbi2d 636	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥)))
53		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝐹‘𝑦) = (𝐹‘∩ 𝑧))
54	53	sseq1d 3946	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥)))
55	52, 54	imbi12d 345	. . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑧 → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
56	55	spcgv 3534	. . . . . . . 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 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ dom (𝐹 ∩ I ))
59	23	3ad2ant1 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝐹 Fn 𝒫 𝐵)
60	59	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝐹 Fn 𝒫 𝐵)
61		fnelfp 7119	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
62	60, 46, 61	syl2anc 590	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
63	58, 62	mpbid 233	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘𝑥) = 𝑥)
64	57, 63	sseqtrd 3951	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ 𝑥)
65	64	ralrimiva 3131	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
66		ssint 4894	. . . . 5 ⊢ ((𝐹‘∩ 𝑧) ⊆ ∩ 𝑧 ↔ ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
67	65, 66	sylibr 235	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) ⊆ ∩ 𝑧)
68	16	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
69		id 22	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → 𝑥 = ∩ 𝑧)
70		fveq2 6827	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → (𝐹‘𝑥) = (𝐹‘∩ 𝑧))
71	69, 70	sseq12d 3948	. . . . . 6 ⊢ (𝑥 = ∩ 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧)))
72	71	rspcva 3558	. . . . 5 ⊢ ((∩ 𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
73	39, 68, 72	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
74	67, 73	eqssd 3932	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) = ∩ 𝑧)
75		fnelfp 7119	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∩ 𝑧 ∈ 𝒫 𝐵) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
76	59, 39, 75	syl2anc 590	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
77	74, 76	mpbird 258	. 2 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ dom (𝐹 ∩ I ))
78	5, 26, 77	ismred 17555	1 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838  ∩ cint 4877   I cid 5512  dom cdm 5618   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  Moorecmre 17535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-mre 17539

This theorem is referenced by:  ismrcd2  43148  istopclsd  43149  ismrc  43150