Theorem isnacs3 42700

Description: A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Assertion

Ref	Expression
isnacs3	⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))

Distinct variable groups:   𝐶,𝑠   𝑋,𝑠

Proof of Theorem isnacs3

Dummy variables 𝑔 ℎ 𝑖 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nacsacs 42699	. . . 4 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
2	1	acsmred 17673	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3		simpll 766	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
4	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5		elpwi 4587	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐶 → 𝑠 ⊆ 𝐶)
6	5	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠 ⊆ 𝐶)
7		simpr 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (toInc‘𝑠) ∈ Dirset)
8		acsdrsel 18558	. . . . . . . . 9 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠 ⊆ 𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
9	4, 6, 7, 8	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
10		eqid 2736	. . . . . . . . 9 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
11	10	nacsfg 42695	. . . . . . . 8 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ ∪ 𝑠 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
12	3, 9, 11	syl2anc 584	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
13	10	mrefg2 42697	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)))
14	2, 13	syl 17	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)))
15	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)))
16	12, 15	mpbid 232	. . . . . 6 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
17		elfpw 9371	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) ↔ (𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin))
18		fissuni 9374	. . . . . . . . 9 ⊢ ((𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin) → ∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ)
19	17, 18	sylbi 217	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) → ∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ)
20		elfpw 9371	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝒫 𝑠 ∩ Fin) ↔ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin))
21		ipodrsfi 18554	. . . . . . . . . . . . 13 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
22	21	3expb 1120	. . . . . . . . . . . 12 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
23	20, 22	sylan2b 594	. . . . . . . . . . 11 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
24		sstr 3972	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ⊆ ∪ ℎ ∧ ∪ ℎ ⊆ 𝑖) → 𝑔 ⊆ 𝑖)
25	24	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ) → 𝑔 ⊆ 𝑖)
26		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
27	2	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑔 ⊆ 𝑖)
29	5	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑠 ⊆ 𝐶)
30		simprl 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝑠)
31	29, 30	sseldd 3964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝐶)
32	10	mrcsscl 17637	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑖 ∧ 𝑖 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
33	27, 28, 31, 32	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
35	26, 34	eqsstrd 3998	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 ⊆ 𝑖)
36		simplrl 776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 ∈ 𝑠)
37		elssuni 4918	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑠 → 𝑖 ⊆ ∪ 𝑠)
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 ⊆ ∪ 𝑠)
39	35, 38	eqssd 3981	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = 𝑖)
40	39, 36	eqeltrd 2835	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 ∈ 𝑠)
41	40	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))
42	41	expr 456	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → (𝑔 ⊆ 𝑖 → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
43	25, 42	syl5 34	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → ((∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
44	43	expd 415	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → (∪ ℎ ⊆ 𝑖 → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
45	44	rexlimdva 3142	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖 → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
46	23, 45	syl5 34	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (((toInc‘𝑠) ∈ Dirset ∧ ℎ ∈ (𝒫 𝑠 ∩ Fin)) → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
47	46	expdimp 452	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (ℎ ∈ (𝒫 𝑠 ∩ Fin) → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
48	47	rexlimdv 3140	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
49	19, 48	syl5 34	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
50	49	rexlimdv 3140	. . . . . 6 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))
51	16, 50	mpd 15	. . . . 5 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝑠)
52	51	ex 412	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
53	52	ralrimiva 3133	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
54	2, 53	jca 511	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))
55		simpl 482	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (Moore‘𝑋))
56	5	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → 𝑠 ⊆ 𝐶)
57	56	sseld 3962	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (∪ 𝑠 ∈ 𝑠 → ∪ 𝑠 ∈ 𝐶))
58	57	imim2d 57	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠) → ((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
59	58	ralimdva 3153	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
60	59	imp 406	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))
61		isacs3 18565	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
62	55, 60, 61	sylanbrc 583	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (ACS‘𝑋))
63	10	mrcid 17630	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
64	63	adantlr 715	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
65	62	adantr 480	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝐶 ∈ (ACS‘𝑋))
66		mress 17610	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
67	66	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
68	65, 10, 67	acsficld 18566	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
69	64, 68	eqtr3d 2773	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
70	10	mrcf 17626	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋⟶𝐶)
71	70	ffnd 6712	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
72	71	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
73	10	mrcss 17633	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
74	73	3expb 1120	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
75	74	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
76		vex 3468	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
77		fpwipodrs 18555	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ V → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
78	76, 77	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
79		inss1 4217	. . . . . . . . . . . 12 ⊢ (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
80	66	sspwd 4593	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
81	79, 80	sstrid 3975	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82		fvex 6894	. . . . . . . . . . . . 13 ⊢ (mrCls‘𝐶) ∈ V
83		imaexg 7914	. . . . . . . . . . . . 13 ⊢ ((mrCls‘𝐶) ∈ V → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
84	82, 83	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V
85	84	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
86	72, 75, 78, 81, 85	ipodrsima 18556	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
87	86	adantlr 715	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
88		imassrn 6063	. . . . . . . . . . . . . 14 ⊢ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
89	70	frnd 6719	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
90	88, 89	sstrid 3975	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
91	90	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
92	84	elpw 4584	. . . . . . . . . . . 12 ⊢ (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
93	91, 92	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
94	93	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
95		simplr 768	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
96		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
97	96	eleq1d 2820	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
98		unieq 4899	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ∪ 𝑠 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
99		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → 𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
100	98, 99	eleq12d 2829	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (∪ 𝑠 ∈ 𝑠 ↔ ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
101	97, 100	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠) ↔ ((toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset → ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))))
102	101	rspcva 3604	. . . . . . . . . 10 ⊢ ((((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → ((toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset → ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
103	94, 95, 102	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset → ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
104	87, 103	mpd 15	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
105	69, 104	eqeltrd 2835	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106		fvelimab 6956	. . . . . . . . 9 ⊢ (((mrCls‘𝐶) Fn 𝒫 𝑋 ∧ (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
107	72, 81, 106	syl2anc 584	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
108	107	adantlr 715	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
109	105, 108	mpbid 232	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
110		eqcom 2743	. . . . . . 7 ⊢ (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111	110	rexbii 3084	. . . . . 6 ⊢ (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
112	109, 111	sylibr 234	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
113	10	mrefg2 42697	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
114	113	ad2antrr 726	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
115	112, 114	mpbird 257	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
116	115	ralrimiva 3133	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → ∀𝑡 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
117	10	isnacs 42694	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
118	62, 116, 117	sylanbrc 583	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
119	54, 118	impbii 209	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4888  ran crn 5660   “ cima 5662   Fn wfn 6531  ‘cfv 6536  Fincfn 8964  Moorecmre 17599  mrClscmrc 17600  ACScacs 17602  Dirsetcdrs 18310  toInccipo 18542  NoeACScnacs 42692

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-tset 17295  df-ple 17296  df-ocomp 17297  df-mre 17603  df-mrc 17604  df-acs 17606  df-proset 18311  df-drs 18312  df-poset 18330  df-ipo 18543  df-nacs 42693

This theorem is referenced by:  nacsfix  42702