Theorem isnacs3 43474

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 43473	. . . 4 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
2	1	acsmred 17730	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3		simpll 779	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
4	1	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5		elpwi 4571	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐶 → 𝑠 ⊆ 𝐶)
6	5	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠 ⊆ 𝐶)
7		simpr 490	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (toInc‘𝑠) ∈ Dirset)
8		acsdrsel 18617	. . . . . . . . 9 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠 ⊆ 𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
9	4, 6, 7, 8	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
10		eqid 2765	. . . . . . . . 9 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
11	10	nacsfg 43469	. . . . . . . 8 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ ∪ 𝑠 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
12	3, 9, 11	syl2anc 596	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
13	10	mrefg2 43471	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)))
14	2, 13	syl 18	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)))
15	14	ad2antrr 739	. . . . . . 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 235	. . . . . 6 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
17		elfpw 9314	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) ↔ (𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin))
18		fissuni 9317	. . . . . . . . 9 ⊢ ((𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin) → ∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ)
19	17, 18	sylbi 220	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) → ∃ℎ ∈ (𝒫 𝑠 ∩ Fin)𝑔 ⊆ ∪ ℎ)
20		elfpw 9314	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝒫 𝑠 ∩ Fin) ↔ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin))
21		ipodrsfi 18613	. . . . . . . . . . . . 13 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
22	21	3expb 1138	. . . . . . . . . . . 12 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
23	20, 22	sylan2b 606	. . . . . . . . . . 11 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
24		sstr 3946	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ⊆ ∪ ℎ ∧ ∪ ℎ ⊆ 𝑖) → 𝑔 ⊆ 𝑖)
25	24	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ) → 𝑔 ⊆ 𝑖)
26		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
27	2	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28		simprr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑔 ⊆ 𝑖)
29	5	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑠 ⊆ 𝐶)
30		simprl 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝑠)
31	29, 30	sseldd 3939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝐶)
32	10	mrcsscl 17694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑖 ∧ 𝑖 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
33	27, 28, 31, 32	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
34	33	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
35	26, 34	eqsstrd 3972	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 ⊆ 𝑖)
36		simplrl 789	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 ∈ 𝑠)
37		elssuni 4906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑠 → 𝑖 ⊆ ∪ 𝑠)
38	36, 37	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 ⊆ ∪ 𝑠)
39	35, 38	eqssd 3955	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = 𝑖)
40	39, 36	eqeltrd 2865	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 ∈ 𝑠)
41	40	ex 418	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))
42	41	expr 462	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → (𝑔 ⊆ 𝑖 → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
43	25, 42	syl5 35	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → ((∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
44	43	expd 421	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖 ∈ 𝑠) → (∪ ℎ ⊆ 𝑖 → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
45	44	rexlimdva 3168	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖 → (𝑔 ⊆ ∪ ℎ → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))))
46	23, 45	syl5 35	. . . . . . . . . 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 458	. . . . . . . . 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 3166	. . . . . . . 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 35	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) → (∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠)))
50	49	rexlimdv 3166	. . . . . 6 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (∃𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔) → ∪ 𝑠 ∈ 𝑠))
51	16, 50	mpd 16	. . . . 5 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝑠)
52	51	ex 418	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
53	52	ralrimiva 3159	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
54	2, 53	jca 521	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))
55		simpl 488	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (Moore‘𝑋))
56	5	adantl 487	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → 𝑠 ⊆ 𝐶)
57	56	sseld 3937	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (∪ 𝑠 ∈ 𝑠 → ∪ 𝑠 ∈ 𝐶))
58	57	imim2d 58	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠) → ((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
59	58	ralimdva 3179	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
60	59	imp 412	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))
61		isacs3 18624	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
62	55, 60, 61	sylanbrc 595	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (ACS‘𝑋))
63	10	mrcid 17687	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
64	63	adantlr 728	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
65	62	adantr 486	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝐶 ∈ (ACS‘𝑋))
66		mress 17663	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
67	66	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
68	65, 10, 67	acsficld 18625	. . . . . . . . 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 2802	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
70	10	mrcf 17683	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋⟶𝐶)
71	70	ffnd 6710	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
72	71	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
73	10	mrcss 17690	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
74	73	3expb 1138	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
75	74	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
76		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
77		fpwipodrs 18614	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ V → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
78	76, 77	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
79		inss1 4189	. . . . . . . . . . . 12 ⊢ (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
80	66	sspwd 4577	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
81	79, 80	sstrid 3949	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82		fvex 6898	. . . . . . . . . . . . 13 ⊢ (mrCls‘𝐶) ∈ V
83		imaexg 7912	. . . . . . . . . . . . 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 18615	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
87	86	adantlr 728	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
88		imassrn 6075	. . . . . . . . . . . . . 14 ⊢ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
89	70	frnd 6718	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
90	88, 89	sstrid 3949	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
91	90	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
92	84	elpw 4568	. . . . . . . . . . . 12 ⊢ (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
93	91, 92	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
94	93	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
95		simplr 781	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
96		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
97	96	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
98		unieq 4885	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ∪ 𝑠 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
99		id 23	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → 𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
100	98, 99	eleq12d 2859	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (∪ 𝑠 ∈ 𝑠 ↔ ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
101	97, 100	imbi12d 347	. . . . . . . . . . 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 3581	. . . . . . . . . 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 596	. . . . . . . . 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 16	. . . . . . . 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 2865	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106		fvelimab 6957	. . . . . . . . 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 596	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
108	107	adantlr 728	. . . . . . 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 235	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
110		eqcom 2772	. . . . . . 7 ⊢ (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111	110	rexbii 3114	. . . . . 6 ⊢ (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
112	109, 111	sylibr 237	. . . . 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 43471	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
114	113	ad2antrr 739	. . . . 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 260	. . . 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 3159	. . 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 43468	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
118	62, 116, 117	sylanbrc 595	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
119	54, 118	impbii 212	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874  ran crn 5664   “ cima 5666   Fn wfn 6535  ‘cfv 6540  Fincfn 8945  Moorecmre 17652  mrClscmrc 17653  ACScacs 17655  Dirsetcdrs 18367  toInccipo 18601  NoeACScnacs 43466

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-struct 17225  df-slot 17260  df-ndx 17272  df-base 17288  df-tset 17347  df-ple 17348  df-ocomp 17349  df-mre 17656  df-mrc 17657  df-acs 17659  df-proset 18368  df-drs 18369  df-poset 18387  df-ipo 18602  df-nacs 43467

This theorem is used by:  nacsfix  43476