isnacs3 - Mathbox for Stefan O'Rear

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Theorem isnacs3 41434

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 41433	. . . 4 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
2	1	acsmred 17597	. . 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 725	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5		elpwi 4609	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐶 → 𝑠 ⊆ 𝐶)
6	5	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠 ⊆ 𝐶)
7		simpr 486	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (toInc‘𝑠) ∈ Dirset)
8		acsdrsel 18493	. . . . . . . . 9 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠 ⊆ 𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
9	4, 6, 7, 8	syl3anc 1372	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
10		eqid 2733	. . . . . . . . 9 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
11	10	nacsfg 41429	. . . . . . . 8 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ ∪ 𝑠 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
12	3, 9, 11	syl2anc 585	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
13	10	mrefg2 41431	. . . . . . . . 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 725	. . . . . . 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 231	. . . . . 6 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
17		elfpw 9351	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) ↔ (𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin))
18		fissuni 9354	. . . . . . . . 9 ⊢ ((𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin) → ∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ)
19	17, 18	sylbi 216	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) → ∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ)
20		elfpw 9351	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝒫 𝑠 ∩ Fin) ↔ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin))
21		ipodrsfi 18489	. . . . . . . . . . . . 13 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
22	21	3expb 1121	. . . . . . . . . . . 12 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
23	20, 22	sylan2b 595	. . . . . . . . . . 11 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
24		sstr 3990	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ⊆ ∪ ℎ ∧ ∪ ℎ ⊆ 𝑖) → 𝑔 ⊆ 𝑖)
25	24	ancoms 460	. . . . . . . . . . . . . 14 ⊢ ((∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ) → 𝑔 ⊆ 𝑖)
26		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
27	2	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝐶)
32	10	mrcsscl 17561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑖 ∧ 𝑖 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
33	27, 28, 31, 32	syl3anc 1372	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
34	33	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
35	26, 34	eqsstrd 4020	. . . . . . . . . . . . . . . . . 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 4941	. . . . . . . . . . . . . . . . . . 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 3999	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = 𝑖)
40	39, 36	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 ∈ 𝑠)
41	40	ex 414	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))
42	41	expr 458	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → (∪ ℎ ⊆ 𝑖 → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
45	44	rexlimdva 3156	. . . . . . . . . . 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 454	. . . . . . . . 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 3154	. . . . . . . 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 3154	. . . . . 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 414	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
53	52	ralrimiva 3147	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
54	2, 53	jca 513	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))
55		simpl 484	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (Moore‘𝑋))
56	5	adantl 483	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → 𝑠 ⊆ 𝐶)
57	56	sseld 3981	. . . . . . 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 3168	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
60	59	imp 408	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))
61		isacs3 18500	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
62	55, 60, 61	sylanbrc 584	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (ACS‘𝑋))
63	10	mrcid 17554	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
64	63	adantlr 714	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
65	62	adantr 482	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝐶 ∈ (ACS‘𝑋))
66		mress 17534	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
67	66	adantlr 714	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
68	65, 10, 67	acsficld 18501	. . . . . . . . 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 2775	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
70	10	mrcf 17550	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋⟶𝐶)
71	70	ffnd 6716	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
72	71	adantr 482	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
73	10	mrcss 17557	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
74	73	3expb 1121	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
75	74	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
76		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
77		fpwipodrs 18490	. . . . . . . . . . . 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 4228	. . . . . . . . . . . 12 ⊢ (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
80	66	sspwd 4615	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
81	79, 80	sstrid 3993	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82		fvex 6902	. . . . . . . . . . . . 13 ⊢ (mrCls‘𝐶) ∈ V
83		imaexg 7903	. . . . . . . . . . . . 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 18491	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
87	86	adantlr 714	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
88		imassrn 6069	. . . . . . . . . . . . . 14 ⊢ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
89	70	frnd 6723	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
90	88, 89	sstrid 3993	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
91	90	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
92	84	elpw 4606	. . . . . . . . . . . 12 ⊢ (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
93	91, 92	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
94	93	adantlr 714	. . . . . . . . . 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 6889	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
97	96	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
98		unieq 4919	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (∪ 𝑠 ∈ 𝑠 ↔ ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
101	97, 100	imbi12d 345	. . . . . . . . . . 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 3611	. . . . . . . . . 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 585	. . . . . . . . 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 2834	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106		fvelimab 6962	. . . . . . . . 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 585	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
108	107	adantlr 714	. . . . . . 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 231	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
110		eqcom 2740	. . . . . . 7 ⊢ (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111	110	rexbii 3095	. . . . . 6 ⊢ (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
112	109, 111	sylibr 233	. . . . 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 41431	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
114	113	ad2antrr 725	. . . . 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 3147	. . 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 41428	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
118	62, 116, 117	sylanbrc 584	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
119	54, 118	impbii 208	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 ran crn 5677 “ cima 5679 Fn wfn 6536 ‘cfv 6541 Fincfn 8936 Moorecmre 17523 mrClscmrc 17524 ACScacs 17526 Dirsetcdrs 18244 toInccipo 18477 NoeACScnacs 41426

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-tset 17213 df-ple 17214 df-ocomp 17215 df-mre 17527 df-mrc 17528 df-acs 17530 df-proset 18245 df-drs 18246 df-poset 18263 df-ipo 18478 df-nacs 41427

This theorem is referenced by: nacsfix 41436

W3C validator