Theorem isnacs3 40448

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 40447	. . . 4 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
2	1	acsmred 17282	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3		simpll 763	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
4	1	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5		elpwi 4539	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐶 → 𝑠 ⊆ 𝐶)
6	5	ad2antlr 723	. . . . . . . . 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 18176	. . . . . . . . 9 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠 ⊆ 𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
9	4, 6, 7, 8	syl3anc 1369	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∪ 𝑠 ∈ 𝐶)
10		eqid 2738	. . . . . . . . 9 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
11	10	nacsfg 40443	. . . . . . . 8 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ ∪ 𝑠 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
12	3, 9, 11	syl2anc 583	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑠 = ((mrCls‘𝐶)‘𝑔))
13	10	mrefg2 40445	. . . . . . . . 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 722	. . . . . . 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 9051	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) ↔ (𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin))
18		fissuni 9054	. . . . . . . . 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 9051	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝒫 𝑠 ∩ Fin) ↔ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin))
21		ipodrsfi 18172	. . . . . . . . . . . . 13 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
22	21	3expb 1118	. . . . . . . . . . . 12 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
23	20, 22	sylan2b 593	. . . . . . . . . . 11 ⊢ (((toInc‘𝑠) ∈ Dirset ∧ ℎ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖)
24		sstr 3925	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28		simprr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑔 ⊆ 𝑖)
29	5	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑠 ⊆ 𝐶)
30		simprl 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝑠)
31	29, 30	sseldd 3918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝐶)
32	10	mrcsscl 17246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ 𝑖 ∧ 𝑖 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
33	27, 28, 31, 32	syl3anc 1369	. . . . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 ⊆ 𝑖)
36		simplrl 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 ∈ 𝑠)
37		elssuni 4868	. . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = 𝑖)
40	39, 36	eqeltrd 2839	. . . . . . . . . . . . . . . 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 3212	. . . . . . . . . . 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 3211	. . . . . . . 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 3211	. . . . . 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 3107	. . 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 3916	. . . . . . 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 3102	. . . . 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 18183	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
62	55, 60, 61	sylanbrc 582	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) → 𝐶 ∈ (ACS‘𝑋))
63	10	mrcid 17239	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
64	63	adantlr 711	. . . . . . . . 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 17219	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
67	66	adantlr 711	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ⊆ 𝑋)
68	65, 10, 67	acsficld 18184	. . . . . . . . 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 2780	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
70	10	mrcf 17235	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋⟶𝐶)
71	70	ffnd 6585	. . . . . . . . . . . 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 17242	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
74	73	3expb 1118	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
75	74	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) ∧ (𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘ℎ))
76		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
77		fpwipodrs 18173	. . . . . . . . . . . 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 4159	. . . . . . . . . . . 12 ⊢ (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
80	66	sspwd 4545	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
81	79, 80	sstrid 3928	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82		fvex 6769	. . . . . . . . . . . . 13 ⊢ (mrCls‘𝐶) ∈ V
83		imaexg 7736	. . . . . . . . . . . . 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 18174	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
87	86	adantlr 711	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
88		imassrn 5969	. . . . . . . . . . . . . 14 ⊢ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
89	70	frnd 6592	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
90	88, 89	sstrid 3928	. . . . . . . . . . . . 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 4534	. . . . . . . . . . . 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 711	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
95		simplr 765	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))
96		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
97	96	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
98		unieq 4847	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 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 3550	. . . . . . . . . 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 583	. . . . . . . . 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 2839	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106		fvelimab 6823	. . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
108	107	adantlr 711	. . . . . . 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 2745	. . . . . . 7 ⊢ (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111	110	rexbii 3177	. . . . . 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 40445	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
114	113	ad2antrr 722	. . . . 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 256	. . . 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 3107	. . 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 40442	. . 3 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
118	62, 116, 117	sylanbrc 582	. 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)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ran crn 5581   “ cima 5583   Fn wfn 6413  ‘cfv 6418  Fincfn 8691  Moorecmre 17208  mrClscmrc 17209  ACScacs 17211  Dirsetcdrs 17927  toInccipo 18160  NoeACScnacs 40440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-slot 16811  df-ndx 16823  df-base 16841  df-tset 16907  df-ple 16908  df-ocomp 16909  df-mre 17212  df-mrc 17213  df-acs 17215  df-proset 17928  df-drs 17929  df-poset 17946  df-ipo 18161  df-nacs 40441

This theorem is referenced by:  nacsfix  40450