Theorem ismrc 40439

Description: A function is a Moore closure operator iff it satisfies mrcssid 17243, mrcss 17242, and mrcidm 17245. (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 17233	. . . . 5 ⊢ mrCls Fn ∪ ran Moore
2		fnfun 6517	. . . . 5 ⊢ (mrCls Fn ∪ ran Moore → Fun mrCls)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun mrCls
4		fvelima 6817	. . . 4 ⊢ ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
5	3, 4	mpan 686	. . 3 ⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6		elfvex 6789	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7		eqid 2738	. . . . . . . 8 ⊢ (mrCls‘𝑧) = (mrCls‘𝑧)
8	7	mrcf 17235	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝑧)
9		mresspw 17218	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
10	8, 9	fssd 6602	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
11	7	mrcssid 17243	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
12	11	adantrr 713	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
13	7	mrcss 17242	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14	13	3expb 1118	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
15	14	ancom2s 646	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
16	7	mrcidm 17245	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
17	16	adantrr 713	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
18	12, 15, 17	3jca 1126	. . . . . . . 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 1932	. . . . . 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 1126	. . . . 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 6565	. . . . . 6 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ↔ 𝐹:𝒫 𝐵⟶𝒫 𝐵))
23		fveq1 6755	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹‘𝑥))
24	23	sseq2d 3949	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹‘𝑥)))
25		fveq1 6755	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹‘𝑦))
26	25, 23	sseq12d 3950	. . . . . . . . 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 6763	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
29	28, 23	eqeq12d 2754	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
30	24, 26, 29	3anbi123d 1434	. . . . . . . 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 1927	. . . . . 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 1437	. . . . 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 244	. . . 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 3208	. . 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 1134	. . . 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 1135	. . . 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 3939	. . . . . . 7 ⊢ 𝑧 ⊆ 𝑧
40		3simpb 1147	. . . . . . . . . . 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 1816	. . . . . . . . 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 3942	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
44	43	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
45		sseq12 3944	. . . . . . . . . . . . . 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 630	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧)))
48		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
49		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
50	48, 49	sseq12d 3950	. . . . . . . . . . . . . 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 6761	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
53	52, 49	eqeq12d 2754	. . . . . . . . . . . . . 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 630	. . . . . . . . . . . 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 3529	. . . . . . . . . 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 3430	. . . . . . . . 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 1133	. . . . . . 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 693	. . . . . 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 1135	. . . . . . . . 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 1816	. . . . . . 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 1133	. . . . . 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 3944	. . . . . . . . . . 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 630	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧)))
72		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
73		sseq12 3944	. . . . . . . . . 10 ⊢ (((𝐹‘𝑦) = (𝐹‘𝑤) ∧ (𝐹‘𝑥) = (𝐹‘𝑧)) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
74	72, 49, 73	syl2anr 596	. . . . . . . . 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 3529	. . . . . . 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 3430	. . . . . 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 1114	. . . 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 40437	. . 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 40436	. . . 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 6785	. . . . . 6 ⊢ (Moore‘𝐵) ⊆ ∪ ran Moore
84	1	fndmi 6521	. . . . . 6 ⊢ dom mrCls = ∪ ran Moore
85	83, 84	sseqtrri 3954	. . . . 5 ⊢ (Moore‘𝐵) ⊆ dom mrCls
86		funfvima2 7089	. . . . 5 ⊢ ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
87	3, 85, 86	mp2an 688	. . . 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 2839	. 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 208	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 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   I cid 5479  dom cdm 5580  ran crn 5581   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  Moorecmre 17208  mrClscmrc 17209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-mre 17212  df-mrc 17213

This theorem is referenced by: (None)