Theorem ismrcd1 37261

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 16277), isotone (satisfies mrcss 16276), and idempotent (satisfies mrcidm 16279) 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 37262 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 3833	. . . 4 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
2		dmss 5323	. . . 4 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . 3 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
4		ismrcd.f	. . . 4 ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
5		fdm 6051	. . . 4 ⊢ (𝐹:𝒫 𝐵⟶𝒫 𝐵 → dom 𝐹 = 𝒫 𝐵)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → dom 𝐹 = 𝒫 𝐵)
7	3, 6	syl5sseq 3653	. 2 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
8		ssid 3624	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
9		ismrcd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
10		elpwg 4166	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
12	8, 11	mpbiri 248	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
13	4, 12	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝒫 𝐵)
14	13	elpwid 4170	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ⊆ 𝐵)
15		selpw 4165	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
16		ismrcd.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
17	15, 16	sylan2b 492	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
18	17	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
19		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
20		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
21	19, 20	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝐵 ⊆ (𝐹‘𝐵)))
22	21	rspcva 3307	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → 𝐵 ⊆ (𝐹‘𝐵))
23	12, 18, 22	syl2anc 693	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐹‘𝐵))
24	14, 23	eqssd 3620	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐵)
25		ffn 6045	. . . . 5 ⊢ (𝐹:𝒫 𝐵⟶𝒫 𝐵 → 𝐹 Fn 𝒫 𝐵)
26	4, 25	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
27		fnelfp 6441	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
28	26, 12, 27	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
29	24, 28	mpbird 247	. 2 ⊢ (𝜑 → 𝐵 ∈ dom (𝐹 ∩ I ))
30		simp2 1062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ dom (𝐹 ∩ I ))
31	7	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
32	30, 31	sstrd 3613	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ 𝒫 𝐵)
33		simp3 1063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ≠ ∅)
34		intssuni2 4502	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝒫 𝐵 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
35	32, 33, 34	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
36		unipw 4918	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐵 = 𝐵
37	35, 36	syl6sseq 3651	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ 𝐵)
38		intex 4820	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ ∅ ↔ ∩ 𝑧 ∈ V)
39		elpwg 4166	. . . . . . . . . . . 12 ⊢ (∩ 𝑧 ∈ V → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
40	38, 39	sylbi 207	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
41	40	3ad2ant3 1084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
42	37, 41	mpbird 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ 𝒫 𝐵)
43	42	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ∈ 𝒫 𝐵)
44		ismrcd.m	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
45	44	3expib 1268	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
46	45	alrimiv 1855	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
47	46	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
48	47	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
49	32	sselda 3603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝒫 𝐵)
50	49	elpwid 4170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ⊆ 𝐵)
51		intss1 4492	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑧 → ∩ 𝑧 ⊆ 𝑥)
52	51	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ⊆ 𝑥)
53	50, 52	jca 554	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥))
54		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝑦 ⊆ 𝑥 ↔ ∩ 𝑧 ⊆ 𝑥))
55	54	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥)))
56		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝐹‘𝑦) = (𝐹‘∩ 𝑧))
57	56	sseq1d 3632	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥)))
58	55, 57	imbi12d 334	. . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑧 → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
59	58	spcgv 3293	. . . . . . . 8 ⊢ (∩ 𝑧 ∈ 𝒫 𝐵 → (∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
60	43, 48, 53, 59	syl3c 66	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))
61	30	sselda 3603	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ dom (𝐹 ∩ I ))
62	26	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝐹 Fn 𝒫 𝐵)
63	62	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝐹 Fn 𝒫 𝐵)
64		fnelfp 6441	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
65	63, 49, 64	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
66	61, 65	mpbid 222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘𝑥) = 𝑥)
67	60, 66	sseqtrd 3641	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ 𝑥)
68	67	ralrimiva 2966	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
69		ssint 4493	. . . . 5 ⊢ ((𝐹‘∩ 𝑧) ⊆ ∩ 𝑧 ↔ ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
70	68, 69	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) ⊆ ∩ 𝑧)
71	18	3ad2ant1 1082	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
72		id 22	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → 𝑥 = ∩ 𝑧)
73		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → (𝐹‘𝑥) = (𝐹‘∩ 𝑧))
74	72, 73	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = ∩ 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧)))
75	74	rspcva 3307	. . . . 5 ⊢ ((∩ 𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
76	42, 71, 75	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
77	70, 76	eqssd 3620	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) = ∩ 𝑧)
78		fnelfp 6441	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∩ 𝑧 ∈ 𝒫 𝐵) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
79	62, 42, 78	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
80	77, 79	mpbird 247	. 2 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ dom (𝐹 ∩ I ))
81	7, 29, 80	ismred 16262	1 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475   I cid 5023  dom cdm 5114   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  Moorecmre 16242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246

This theorem is referenced by:  ismrcd2  37262  istopclsd  37263  ismrc  37264