Theorem ismrc 36282

Description: A function is a Moore closure operator iff it satisfies mrcssid 16100, mrcss 16099, and mrcidm 16102. (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 16090	. . . . 5 ⊢ mrCls Fn ∪ ran Moore
2		fnfun 5902	. . . . 5 ⊢ (mrCls Fn ∪ ran Moore → Fun mrCls)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun mrCls
4		fvelima 6158	. . . 4 ⊢ ((Fun mrCls ∧ 𝐹 ∈ (mrCls “ (Moore‘𝐵))) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
5	3, 4	mpan 702	. . 3 ⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) → ∃𝑧 ∈ (Moore‘𝐵)(mrCls‘𝑧) = 𝐹)
6		elfvex 6131	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝐵 ∈ V)
7		eqid 2610	. . . . . . . 8 ⊢ (mrCls‘𝑧) = (mrCls‘𝑧)
8	7	mrcf 16092	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝑧)
9		mresspw 16075	. . . . . . 7 ⊢ (𝑧 ∈ (Moore‘𝐵) → 𝑧 ⊆ 𝒫 𝐵)
10	8, 9	fssd 5970	. . . . . 6 ⊢ (𝑧 ∈ (Moore‘𝐵) → (mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵)
11	7	mrcssid 16100	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
12	11	adantrr 749	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ((mrCls‘𝑧)‘𝑥))
13	7	mrcss 16099	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
14	13	3expb 1258	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
15	14	ancom2s 840	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘𝑦) ⊆ ((mrCls‘𝑧)‘𝑥))
16	7	mrcidm 16102	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ 𝑥 ⊆ 𝐵) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
17	16	adantrr 749	. . . . . . . . 9 ⊢ ((𝑧 ∈ (Moore‘𝐵) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥)) → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥))
18	12, 15, 17	3jca 1235	. . . . . . . 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 449	. . . . . . 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 1843	. . . . . 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 1235	. . . . 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 5939	. . . . . 6 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧):𝒫 𝐵⟶𝒫 𝐵 ↔ 𝐹:𝒫 𝐵⟶𝒫 𝐵))
23		fveq1 6102	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑥) = (𝐹‘𝑥))
24	23	sseq2d 3596	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (𝑥 ⊆ ((mrCls‘𝑧)‘𝑥) ↔ 𝑥 ⊆ (𝐹‘𝑥)))
25		fveq1 6102	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘𝑦) = (𝐹‘𝑦))
26	25, 23	sseq12d 3597	. . . . . . . . 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 6109	. . . . . . . . . 10 ⊢ ((mrCls‘𝑧) = 𝐹 → ((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
29	28, 23	eqeq12d 2625	. . . . . . . . 9 ⊢ ((mrCls‘𝑧) = 𝐹 → (((mrCls‘𝑧)‘((mrCls‘𝑧)‘𝑥)) = ((mrCls‘𝑧)‘𝑥) ↔ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)))
30	24, 26, 29	3anbi123d 1391	. . . . . . . 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 329	. . . . . . 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 1838	. . . . . 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 1394	. . . . 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 234	. . . 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 3009	. . 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 1054	. . . 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 1055	. . . 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 3587	. . . . . . 7 ⊢ 𝑧 ⊆ 𝑧
40		3simpb 1052	. . . . . . . . . . 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 1731	. . . . . . . . 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		vex 3176	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
44		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
46		sseq12 3591	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧))
47	46	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧))
48	45, 47	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧)))
49		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
50		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
51	49, 50	sseq12d 3597	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
52	51	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑧 ⊆ (𝐹‘𝑧)))
53	50	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
54	53, 50	eqeq12d 2625	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
55	54	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧)))
56	52, 55	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ↔ (𝑧 ⊆ (𝐹‘𝑧) ∧ (𝐹‘(𝐹‘𝑧)) = (𝐹‘𝑧))))
57	48, 56	imbi12d 333	. . . . . . . . . . 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)))))
58	57	spc2gv 3269	. . . . . . . . . 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)))))
59	43, 43, 58	mp2an 704	. . . . . . . . 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))))
60	42, 59	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))))
61	60	3ad2ant3 1077	. . . . . . 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))))
62	39, 61	mpan2i 709	. . . . . 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))))
63	62	imp 444	. . . . 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)))
64	63	simpld 474	. . . 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))
65		simp2 1055	. . . . . . . . 9 ⊢ ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
66	65	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)))
67	66	2alimi 1731	. . . . . . 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)))
68	67	3ad2ant3 1077	. . . . . 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)))
69		vex 3176	. . . . . . 7 ⊢ 𝑤 ∈ V
70	44	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵))
71		sseq12 3591	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) → (𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧))
72	71	ancoms 468	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧))
73	70, 72	anbi12d 743	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧)))
74		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
75		sseq12 3591	. . . . . . . . . 10 ⊢ (((𝐹‘𝑦) = (𝐹‘𝑤) ∧ (𝐹‘𝑥) = (𝐹‘𝑧)) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
76	74, 50, 75	syl2anr 494	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
77	73, 76	imbi12d 333	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))))
78	77	spc2gv 3269	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧))))
79	43, 69, 78	mp2an 704	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧) → (𝐹‘𝑤) ⊆ (𝐹‘𝑧)))
80	68, 79	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)))
81	80	3impib 1254	. . . 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))
82	63	simprd 478	. . . 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))
83	37, 38, 64, 81, 82	ismrcd2 36280	. . 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 )))
84	37, 38, 64, 81, 82	ismrcd1 36279	. . . 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))
85		fvssunirn 6127	. . . . . 6 ⊢ (Moore‘𝐵) ⊆ ∪ ran Moore
86		fndm 5904	. . . . . . 7 ⊢ (mrCls Fn ∪ ran Moore → dom mrCls = ∪ ran Moore)
87	1, 86	ax-mp 5	. . . . . 6 ⊢ dom mrCls = ∪ ran Moore
88	85, 87	sseqtr4i 3601	. . . . 5 ⊢ (Moore‘𝐵) ⊆ dom mrCls
89		funfvima2 6397	. . . . 5 ⊢ ((Fun mrCls ∧ (Moore‘𝐵) ⊆ dom mrCls) → (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵))))
90	3, 88, 89	mp2an 704	. . . 4 ⊢ (dom (𝐹 ∩ I ) ∈ (Moore‘𝐵) → (mrCls‘dom (𝐹 ∩ I )) ∈ (mrCls “ (Moore‘𝐵)))
91	84, 90	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)))
92	83, 91	eqeltrd 2688	. 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)))
93	36, 92	impbii 198	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 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   I cid 4948  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Moorecmre 16065  mrClscmrc 16066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070

This theorem is referenced by: (None)