Theorem ismrc 43711

Description: A function is a Moore closure operator iff it satisfies mrcssid 17791, mrcss 17790, and mrcidm 17793. (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 17781	. . . . 5 ⊢ mrCls Fn ∪ ran Moore
2		fnfun 6639	. . . . 5 ⊢ (mrCls Fn ∪ ran Moore → Fun mrCls)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun mrCls
4		fvelima 6950	. . . 4 ⊢ ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
5	3, 4	mpan 703	. . 3 ⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6		elfvex 6920	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7		eqid 2761	. . . . . . . 8 ⊢ (mrCls‘𝑧) = (mrCls‘𝑧)
8	7	mrcf 17783	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝑧)
9		mresspw 17762	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
10	8, 9	fssd 6727	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
11	7	mrcssid 17791	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
12	11	adantrr 730	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
13	7	mrcss 17790	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14	13	3expb 1138	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
15	14	ancom2s 663	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
16	7	mrcidm 17793	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
17	16	adantrr 730	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
18	12, 15, 17	3jca 1146	. . . . . . . 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 418	. . . . . . 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 1961	. . . . . 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 1146	. . . . 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 6687	. . . . . 6 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ↔ 𝐹:𝒫 𝐵⟶𝒫 𝐵))
23		fveq1 6884	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹‘𝑥))
24	23	sseq2d 3963	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹‘𝑥)))
25		fveq1 6884	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹‘𝑦))
26	25, 23	sseq12d 3964	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
27		id 23	. . . . . . . . . . 11 ⊢ ((mrCls‘𝑧) = 𝐹 → (mrCls‘𝑧) = 𝐹)
28	27, 23	fveq12d 6892	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
29	28, 23	eqeq12d 2777	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
30	24, 26, 29	3anbi123d 1464	. . . . . . . 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 343	. . . . . . 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 1956	. . . . . 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 1467	. . . . 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 248	. . . 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 3157	. . 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 18	. 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 1154	. . . 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 1155	. . . 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 3953	. . . . . . 7 ⊢ 𝑧 ⊆ 𝑧
40		3simpb 1167	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
41	40	imim2i 17	. . . . . . . . . 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 1845	. . . . . . . . 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 3956	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
44	43	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
45		sseq12 3958	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧))
46	45	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧))
47	44, 46	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧)))
48		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
49		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
50	48, 49	sseq12d 3964	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
51	50	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
52		2fveq3 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
53	52, 49	eqeq12d 2777	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
54	53	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
55	51, 54	anbi12d 644	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 3555	. . . . . . . . . 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 3458	. . . . . . . . 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 18	. . . . . . . 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 1153	. . . . . . 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 710	. . . . . 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 412	. . . . 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 500	. . . 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 1155	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
65	64	imim2i 17	. . . . . . . 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 1845	. . . . . . 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 1153	. . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
69		sseq12 3958	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧))
70	69	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧))
71	68, 70	anbi12d 644	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧)))
72		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
73		sseq12 3958	. . . . . . . . . 10 ⊢ (((𝐹‘𝑦) = (𝐹‘𝑤) ∧ (𝐹‘𝑥) = (𝐹‘𝑧)) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
74	72, 49, 73	syl2anr 609	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
75	71, 74	imbi12d 347	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))))
76	75	spc2gv 3555	. . . . . . 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 3458	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
78	67, 77	syl 18	. . . . 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 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)))) ∧ 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))
80	62	simprd 501	. . . 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 43709	. . 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 43708	. . . 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 6916	. . . . . 6 ⊢ (Moore‘𝐵) ⊆ ∪ ran Moore
84	1	fndmi 6643	. . . . . 6 ⊢ dom mrCls = ∪ ran Moore
85	83, 84	sseqtrri 3980	. . . . 5 ⊢ (Moore‘𝐵) ⊆ dom mrCls
86		funfvima2 7237	. . . . 5 ⊢ ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
87	3, 85, 86	mp2an 705	. . . 4 ⊢ (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
88	82, 87	syl 18	. . 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 2861	. 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 212	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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   I cid 5545  dom cdm 5651  ran crn 5652   “ cima 5654  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Moorecmre 17752  mrClscmrc 17753

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-mre 17756  df-mrc 17757

This theorem is used by: (None)