Theorem ismrc 42662

Description: A function is a Moore closure operator iff it satisfies mrcssid 17554, mrcss 17553, and mrcidm 17556. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
ismrc	⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))

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

Proof of Theorem ismrc

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmrc 17544	. . . . 5 ⊢ mrCls Fn ∪ ran Moore
2		fnfun 6600	. . . . 5 ⊢ (mrCls Fn ∪ ran Moore → Fun mrCls)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun mrCls
4		fvelima 6908	. . . 4 ⊢ ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
5	3, 4	mpan 690	. . 3 ⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6		elfvex 6878	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7		eqid 2729	. . . . . . . 8 ⊢ (mrCls‘𝑧) = (mrCls‘𝑧)
8	7	mrcf 17546	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝑧)
9		mresspw 17529	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
10	8, 9	fssd 6687	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
11	7	mrcssid 17554	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
12	11	adantrr 717	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
13	7	mrcss 17553	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14	13	3expb 1120	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
15	14	ancom2s 650	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
16	7	mrcidm 17556	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
17	16	adantrr 717	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
18	12, 15, 17	3jca 1128	. . . . . . . 8 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))
19	18	ex 412	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
20	19	alrimivv 1928	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))))
21	6, 10, 20	3jca 1128	. . . . 5 ⊢ (𝑧 ∈ (Moore‘𝐵) → (𝐵 ∈ V ∧ (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))))
22		feq1 6648	. . . . . 6 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ↔ 𝐹:𝒫 𝐵⟶𝒫 𝐵))
23		fveq1 6839	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹‘𝑥))
24	23	sseq2d 3976	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹‘𝑥)))
25		fveq1 6839	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹‘𝑦))
26	25, 23	sseq12d 3977	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
27		id 22	. . . . . . . . . . 11 ⊢ ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
28	27, 23	fveq12d 6847	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
29	28, 23	eqeq12d 2745	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
30	24, 26, 29	3anbi123d 1438	. . . . . . . 8 ⊢ ((mrCls‘𝑧) = 𝐹 → ((𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)) ↔ (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))
31	30	imbi2d 340	. . . . . . 7 ⊢ ((mrCls‘𝑧) = 𝐹 → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))) ↔ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))
32	31	2albidv 1923	. . . . . 6 ⊢ ((mrCls‘𝑧) = 𝐹 → (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))) ↔ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))
33	22, 32	3anbi23d 1441	. . . . 5 ⊢ ((mrCls‘𝑧) = 𝐹 → ((𝐵 ∈ V ∧ (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ∧ ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥)))) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))))
34	21, 33	syl5ibcom 245	. . . 4 ⊢ (𝑧 ∈ (Moore‘𝐵) → ((mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))))
35	34	rexlimiv 3127	. . 3 ⊢ (∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹 → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))
36	5, 35	syl 17	. 2 ⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))
37		simp1 1136	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → 𝐵 ∈ V)
38		simp2 1137	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
39		ssid 3966	. . . . . . 7 ⊢ 𝑧 ⊆ 𝑧
40		3simpb 1149	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
41	40	imim2i 16	. . . . . . . . . 10 ⊢ (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))
42	41	2alimi 1812	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))
43		sseq1 3969	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
44	43	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
45		sseq12 3971	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧))
46	45	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧))
47	44, 46	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧)))
48		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
49		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
50	48, 49	sseq12d 3977	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
51	50	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
52		2fveq3 6845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
53	52, 49	eqeq12d 2745	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
54	53	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
55	51, 54	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ↔ (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
56	47, 55	imbi12d 344	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) ↔ ((𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧) → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))))
57	56	spc2gv 3563	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ 𝑧 ∈ V) → (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧) → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))))
58	57	el2v 3451	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧) → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
59	42, 58	syl 17	. . . . . . . 8 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧) → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
60	59	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧) → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
61	39, 60	mpan2i 697	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → (𝑧 ⊆ 𝐵 → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
62	61	imp 406	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) ∧ 𝑧 ⊆ 𝐵) → (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
63	62	simpld 494	. . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ (𝐹‘𝑧))
64		simp2 1137	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
65	64	imim2i 16	. . . . . . . 8 ⊢ (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
66	65	2alimi 1812	. . . . . . 7 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))) → ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
67	66	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
68	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
69		sseq12 3971	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧))
70	69	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧))
71	68, 70	anbi12d 632	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧)))
72		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
73		sseq12 3971	. . . . . . . . . 10 ⊢ (((𝐹‘𝑦) = (𝐹‘𝑤) ∧ (𝐹‘𝑥) = (𝐹‘𝑧)) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
74	72, 49, 73	syl2anr 597	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
75	71, 74	imbi12d 344	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))))
76	75	spc2gv 3563	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))))
77	76	el2v 3451	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
78	67, 77	syl 17	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
79	78	3impib 1116	. . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) ∧ 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))
80	62	simprd 495	. . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) ∧ 𝑧 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))
81	37, 38, 63, 79, 80	ismrcd2 42660	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
82	37, 38, 63, 79, 80	ismrcd1 42659	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
83		fvssunirn 6873	. . . . . 6 ⊢ (Moore‘𝐵) ⊆ ∪ ran Moore
84	1	fndmi 6604	. . . . . 6 ⊢ dom mrCls = ∪ ran Moore
85	83, 84	sseqtrri 3993	. . . . 5 ⊢ (Moore‘𝐵) ⊆ dom mrCls
86		funfvima2 7187	. . . . 5 ⊢ ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
87	3, 85, 86	mp2an 692	. . . 4 ⊢ (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
88	82, 87	syl 17	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
89	81, 88	eqeltrd 2828	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))) → 𝐹 ∈ (mrCls “ (Moore‘𝐵)))
90	36, 89	impbii 209	1 ⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   I cid 5525  dom cdm 5631  ran crn 5632   “ cima 5634  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  Moorecmre 17519  mrClscmrc 17520

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  ax-un 7691

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-sbc 3751  df-csb 3860  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-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-mre 17523  df-mrc 17524

This theorem is referenced by: (None)