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Theorem isnacs3 41911
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 → 𝑠𝑠)))
Distinct variable groups:   𝐶,𝑠   𝑋,𝑠

Proof of Theorem isnacs3
Dummy variables 𝑔 𝑖 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nacsacs 41910 . . . 4 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
21acsmred 17607 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3 simpll 764 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
41ad2antrr 723 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5 elpwi 4609 . . . . . . . . . 10 (𝑠 ∈ 𝒫 𝐶𝑠𝐶)
65ad2antlr 724 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
7 simpr 484 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (toInc‘𝑠) ∈ Dirset)
8 acsdrsel 18506 . . . . . . . . 9 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
94, 6, 7, 8syl3anc 1370 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
10 eqid 2731 . . . . . . . . 9 (mrCls‘𝐶) = (mrCls‘𝐶)
1110nacsfg 41906 . . . . . . . 8 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
123, 9, 11syl2anc 583 . . . . . . 7 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
1310mrefg2 41908 . . . . . . . . 9 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔)))
142, 13syl 17 . . . . . . . 8 (𝐶 ∈ (NoeACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔)))
1514ad2antrr 723 . . . . . . 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)))
1612, 15mpbid 231 . . . . . 6 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
17 elfpw 9360 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑔 𝑠𝑔 ∈ Fin))
18 fissuni 9363 . . . . . . . . 9 ((𝑔 𝑠𝑔 ∈ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
1917, 18sylbi 216 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
20 elfpw 9360 . . . . . . . . . . . 12 ( ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑠 ∈ Fin))
21 ipodrsfi 18502 . . . . . . . . . . . . 13 (((toInc‘𝑠) ∈ Dirset ∧ 𝑠 ∈ Fin) → ∃𝑖𝑠 𝑖)
22213expb 1119 . . . . . . . . . . . 12 (((toInc‘𝑠) ∈ Dirset ∧ (𝑠 ∈ Fin)) → ∃𝑖𝑠 𝑖)
2320, 22sylan2b 593 . . . . . . . . . . 11 (((toInc‘𝑠) ∈ Dirset ∧ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖𝑠 𝑖)
24 sstr 3990 . . . . . . . . . . . . . . 15 ((𝑔 𝑖) → 𝑔𝑖)
2524ancoms 458 . . . . . . . . . . . . . 14 (( 𝑖𝑔 ) → 𝑔𝑖)
26 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = ((mrCls‘𝐶)‘𝑔))
272ad2antrr 723 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28 simprr 770 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑔𝑖)
295ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑠𝐶)
30 simprl 768 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝑠)
3129, 30sseldd 3983 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝐶)
3210mrcsscl 17571 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑖𝑖𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3327, 28, 31, 32syl3anc 1370 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3433adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3526, 34eqsstrd 4020 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑖)
36 simplrl 774 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖𝑠)
37 elssuni 4941 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑠𝑖 𝑠)
3836, 37syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 𝑠)
3935, 38eqssd 3999 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = 𝑖)
4039, 36eqeltrd 2832 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑠)
4140ex 412 . . . . . . . . . . . . . . 15 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))
4241expr 456 . . . . . . . . . . . . . 14 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖𝑠) → (𝑔𝑖 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠)))
4325, 42syl5 34 . . . . . . . . . . . . 13 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖𝑠) → (( 𝑖𝑔 ) → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠)))
4443expd 415 . . . . . . . . . . . 12 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖𝑠) → ( 𝑖 → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4544rexlimdva 3154 . . . . . . . . . . 11 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (∃𝑖𝑠 𝑖 → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4623, 45syl5 34 . . . . . . . . . 10 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (((toInc‘𝑠) ∈ Dirset ∧ ∈ (𝒫 𝑠 ∩ Fin)) → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4746expdimp 452 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ( ∈ (𝒫 𝑠 ∩ Fin) → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4847rexlimdv 3152 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠)))
4919, 48syl5 34 . . . . . . 7 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (𝑔 ∈ (𝒫 𝑠 ∩ Fin) → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠)))
5049rexlimdv 3152 . . . . . 6 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))
5116, 50mpd 15 . . . . 5 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝑠)
5251ex 412 . . . 4 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
5352ralrimiva 3145 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
542, 53jca 511 . 2 (𝐶 ∈ (NoeACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)))
55 simpl 482 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (Moore‘𝑋))
565adantl 481 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → 𝑠𝐶)
5756sseld 3981 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ( 𝑠𝑠 𝑠𝐶))
5857imim2d 57 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → (((toInc‘𝑠) ∈ Dirset → 𝑠𝑠) → ((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
5958ralimdva 3166 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
6059imp 406 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶))
61 isacs3 18513 . . . 4 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
6255, 60, 61sylanbrc 582 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (ACS‘𝑋))
6310mrcid 17564 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
6463adantlr 712 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
6562adantr 480 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝐶 ∈ (ACS‘𝑋))
66 mress 17544 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝑡𝑋)
6766adantlr 712 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡𝑋)
6865, 10, 67acsficld 18514 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
6964, 68eqtr3d 2773 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
7010mrcf 17560 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋𝐶)
7170ffnd 6718 . . . . . . . . . . . 12 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7271adantr 480 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7310mrcss 17567 . . . . . . . . . . . . 13 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
74733expb 1119 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
7574adantlr 712 . . . . . . . . . . 11 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
76 vex 3477 . . . . . . . . . . . 12 𝑡 ∈ V
77 fpwipodrs 18503 . . . . . . . . . . . 12 (𝑡 ∈ V → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
7876, 77mp1i 13 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
79 inss1 4228 . . . . . . . . . . . 12 (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
8066sspwd 4615 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
8179, 80sstrid 3993 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82 fvex 6904 . . . . . . . . . . . . 13 (mrCls‘𝐶) ∈ V
83 imaexg 7910 . . . . . . . . . . . . 13 ((mrCls‘𝐶) ∈ V → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
8482, 83ax-mp 5 . . . . . . . . . . . 12 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V
8584a1i 11 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
8672, 75, 78, 81, 85ipodrsima 18504 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
8786adantlr 712 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
88 imassrn 6070 . . . . . . . . . . . . . 14 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
8970frnd 6725 . . . . . . . . . . . . . 14 (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
9088, 89sstrid 3993 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9190adantr 480 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9284elpw 4606 . . . . . . . . . . . 12 (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9391, 92sylibr 233 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
9493adantlr 712 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
95 simplr 766 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
96 fveq2 6891 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
9796eleq1d 2817 . . . . . . . . . . . 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)))
10098, 99eleq12d 2826 . . . . . . . . . . . 12 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ( 𝑠𝑠 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
10197, 100imbi12d 344 . . . . . . . . . . 11 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (((toInc‘𝑠) ∈ Dirset → 𝑠𝑠) ↔ ((toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))))
102101rspcva 3610 . . . . . . . . . 10 ((((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ((toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
10394, 95, 102syl2anc 583 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
10487, 103mpd 15 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
10569, 104eqeltrd 2832 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106 fvelimab 6964 . . . . . . . . 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))
10772, 81, 106syl2anc 583 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
108107adantlr 712 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
109105, 108mpbid 231 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
110 eqcom 2738 . . . . . . 7 (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111110rexbii 3093 . . . . . 6 (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
112109, 111sylibr 233 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11310mrefg2 41908 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
114113ad2antrr 723 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
115112, 114mpbird 257 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
116115ralrimiva 3145 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11710isnacs 41905 . . 3 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
11862, 116, 117sylanbrc 582 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
11954, 118impbii 208 1 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  wrex 3069  Vcvv 3473  cin 3947  wss 3948  𝒫 cpw 4602   cuni 4908  ran crn 5677  cima 5679   Fn wfn 6538  cfv 6543  Fincfn 8945  Moorecmre 17533  mrClscmrc 17534  ACScacs 17536  Dirsetcdrs 18257  toInccipo 18490  NoeACScnacs 41903
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-tset 17223  df-ple 17224  df-ocomp 17225  df-mre 17537  df-mrc 17538  df-acs 17540  df-proset 18258  df-drs 18259  df-poset 18276  df-ipo 18491  df-nacs 41904
This theorem is referenced by:  nacsfix  41913
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