Theorem ismrcd1 41007

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17497), isotone (satisfies mrcss 17496), and idempotent (satisfies mrcidm 17499) 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 41008 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 4188	. . . 4 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
2		dmss 5858	. . . 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 6688	. 2 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
6		ssid 3966	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
7		ismrcd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8		elpwg 4563	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
10	6, 9	mpbiri 257	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
11	4, 10	ffvelcdmd 7036	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝒫 𝐵)
12	11	elpwid 4569	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ⊆ 𝐵)
13		velpw 4565	. . . . . . 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 3143	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
19	17, 18	sseq12d 3977	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝐵 ⊆ (𝐹‘𝐵)))
20	19	rspcva 3579	. . . . 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 6669	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
24		fnelfp 7121	. . . 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 256	. 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 4934	. . . . . . . . . . . 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 5407	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐵 = 𝐵
34	32, 33	sseqtrdi 3994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ 𝐵)
35		intex 5294	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ ∅ ↔ ∩ 𝑧 ∈ V)
36		elpwg 4563	. . . . . . . . . . . 12 ⊢ (∩ 𝑧 ∈ V → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
37	35, 36	sylbi 216	. . . . . . . . . . 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 256	. . . . . . . . 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 1122	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
43	42	alrimiv 1930	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
46	29	sselda 3944	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝒫 𝐵)
47	46	elpwid 4569	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ⊆ 𝐵)
48		intss1 4924	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑧 → ∩ 𝑧 ⊆ 𝑥)
49	48	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ⊆ 𝑥)
50	47, 49	jca 512	. . . . . . . 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 629	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥)))
53		fveq2 6842	. . . . . . . . . . 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 3555	. . . . . . . 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 3944	. . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝐹 Fn 𝒫 𝐵)
61		fnelfp 7121	. . . . . . . . 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 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘𝑥) = 𝑥)
64	57, 63	sseqtrd 3984	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ 𝑥)
65	64	ralrimiva 3143	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
66		ssint 4925	. . . . 5 ⊢ ((𝐹‘∩ 𝑧) ⊆ ∩ 𝑧 ↔ ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
67	65, 66	sylibr 233	. . . 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 6842	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → (𝐹‘𝑥) = (𝐹‘∩ 𝑧))
71	69, 70	sseq12d 3977	. . . . . 6 ⊢ (𝑥 = ∩ 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧)))
72	71	rspcva 3579	. . . . 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 7121	. . . 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 256	. 2 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ dom (𝐹 ∩ I ))
78	5, 26, 77	ismred 17482	1 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4865  ∩ cint 4907   I cid 5530  dom cdm 5633   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  Moorecmre 17462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-mre 17466

This theorem is referenced by:  ismrcd2  41008  istopclsd  41009  ismrc  41010