Theorem isnacs3 42743

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 42742	. . . 4 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
2	1	acsmred 17557	. . 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 4552	. . . . . . . . . 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 18444	. . . . . . . . 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 2731	. . . . . . . . 9 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
11	10	nacsfg 42738	. . . . . . . 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 42740	. . . . . . . . 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 9233	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 ∪ 𝑠 ∩ Fin) ↔ (𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin))
18		fissuni 9236	. . . . . . . . 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 9233	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝒫 𝑠 ∩ Fin) ↔ (ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin))
21		ipodrsfi 18440	. . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) → 𝑖 ∈ 𝐶)
32	10	mrcsscl 17521	. . . . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . 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 4884	. . . . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖)) ∧ ∪ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ∪ 𝑠 = 𝑖)
40	39, 36	eqeltrd 2831	. . . . . . . . . . . . . . . 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 3133	. . . . . . . . . . 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 3131	. . . . . . . 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 3131	. . . . . 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 3124	. . 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 3928	. . . . . . 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 3144	. . . . 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 18451	. . . 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 17514	. . . . . . . . . 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 17490	. . . . . . . . . . 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 18452	. . . . . . . . 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 2768	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 = ∪ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
70	10	mrcf 17510	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋⟶𝐶)
71	70	ffnd 6647	. . . . . . . . . . . 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 17517	. . . . . . . . . . . . 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 3440	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
77		fpwipodrs 18441	. . . . . . . . . . . 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 4182	. . . . . . . . . . . 12 ⊢ (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
80	66	sspwd 4558	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
81	79, 80	sstrid 3941	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ∈ 𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82		fvex 6830	. . . . . . . . . . . . 13 ⊢ (mrCls‘𝐶) ∈ V
83		imaexg 7838	. . . . . . . . . . . . 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 18442	. . . . . . . . . 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 6015	. . . . . . . . . . . . . 14 ⊢ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
89	70	frnd 6654	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
90	88, 89	sstrid 3941	. . . . . . . . . . . . 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 4549	. . . . . . . . . . . 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 6817	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
97	96	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
98		unieq 4865	. . . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 3570	. . . . . . . . . 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 2831	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠)) ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106		fvelimab 6889	. . . . . . . . 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 2738	. . . . . . 7 ⊢ (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111	110	rexbii 3079	. . . . . 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 42740	. . . . . 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 3124	. . 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 42737	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4545  ∪ cuni 4854  ran crn 5612   “ cima 5614   Fn wfn 6471  ‘cfv 6476  Fincfn 8864  Moorecmre 17479  mrClscmrc 17480  ACScacs 17482  Dirsetcdrs 18194  toInccipo 18428  NoeACScnacs 42735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-struct 17053  df-slot 17088  df-ndx 17100  df-base 17116  df-tset 17175  df-ple 17176  df-ocomp 17177  df-mre 17483  df-mrc 17484  df-acs 17486  df-proset 18195  df-drs 18196  df-poset 18214  df-ipo 18429  df-nacs 42736

This theorem is referenced by:  nacsfix  42745