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Theorem isnacs3 38742
 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 38741 . . . 4 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
21acsmred 16744 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3 simpll 763 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
41ad2antrr 722 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5 elpwi 4457 . . . . . . . . . 10 (𝑠 ∈ 𝒫 𝐶𝑠𝐶)
65ad2antlr 723 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
7 simpr 485 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (toInc‘𝑠) ∈ Dirset)
8 acsdrsel 17594 . . . . . . . . 9 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
94, 6, 7, 8syl3anc 1362 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
10 eqid 2793 . . . . . . . . 9 (mrCls‘𝐶) = (mrCls‘𝐶)
1110nacsfg 38737 . . . . . . . 8 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
123, 9, 11syl2anc 584 . . . . . . 7 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
1310mrefg2 38739 . . . . . . . . 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 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)))
1612, 15mpbid 233 . . . . . 6 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
17 elfpw 8662 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑔 𝑠𝑔 ∈ Fin))
18 fissuni 8665 . . . . . . . . 9 ((𝑔 𝑠𝑔 ∈ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
1917, 18sylbi 218 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
20 elfpw 8662 . . . . . . . . . . . 12 ( ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑠 ∈ Fin))
21 ipodrsfi 17590 . . . . . . . . . . . . 13 (((toInc‘𝑠) ∈ Dirset ∧ 𝑠 ∈ Fin) → ∃𝑖𝑠 𝑖)
22213expb 1111 . . . . . . . . . . . 12 (((toInc‘𝑠) ∈ Dirset ∧ (𝑠 ∈ Fin)) → ∃𝑖𝑠 𝑖)
2320, 22sylan2b 593 . . . . . . . . . . 11 (((toInc‘𝑠) ∈ Dirset ∧ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖𝑠 𝑖)
24 sstr 3892 . . . . . . . . . . . . . . 15 ((𝑔 𝑖) → 𝑔𝑖)
2524ancoms 459 . . . . . . . . . . . . . 14 (( 𝑖𝑔 ) → 𝑔𝑖)
26 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = ((mrCls‘𝐶)‘𝑔))
272ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28 simprr 769 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑔𝑖)
295ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑠𝐶)
30 simprl 767 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝑠)
3129, 30sseldd 3885 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝐶)
3210mrcsscl 16708 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑖𝑖𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3327, 28, 31, 32syl3anc 1362 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3433adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3526, 34eqsstrd 3921 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑖)
36 simplrl 773 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖𝑠)
37 elssuni 4768 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑠𝑖 𝑠)
3836, 37syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 𝑠)
3935, 38eqssd 3901 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = 𝑖)
4039, 36eqeltrd 2881 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑠)
4140ex 413 . . . . . . . . . . . . . . 15 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))
4241expr 457 . . . . . . . . . . . . . 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 416 . . . . . . . . . . . 12 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖𝑠) → ( 𝑖 → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4544rexlimdva 3244 . . . . . . . . . . 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 453 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ( ∈ (𝒫 𝑠 ∩ Fin) → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4847rexlimdv 3243 . . . . . . . 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 3243 . . . . . 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 413 . . . 4 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
5352ralrimiva 3147 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
542, 53jca 512 . 2 (𝐶 ∈ (NoeACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)))
55 simpl 483 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (Moore‘𝑋))
565adantl 482 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → 𝑠𝐶)
5756sseld 3883 . . . . . . 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 3142 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
6059imp 407 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶))
61 isacs3 17601 . . . 4 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
6255, 60, 61sylanbrc 583 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (ACS‘𝑋))
6310mrcid 16701 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
6463adantlr 711 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
6562adantr 481 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝐶 ∈ (ACS‘𝑋))
66 mress 16681 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝑡𝑋)
6766adantlr 711 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡𝑋)
6865, 10, 67acsficld 17602 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
6964, 68eqtr3d 2831 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
7010mrcf 16697 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋𝐶)
7170ffnd 6375 . . . . . . . . . . . 12 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7271adantr 481 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7310mrcss 16704 . . . . . . . . . . . . 13 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
74733expb 1111 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
7574adantlr 711 . . . . . . . . . . 11 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
76 vex 3435 . . . . . . . . . . . 12 𝑡 ∈ V
77 fpwipodrs 17591 . . . . . . . . . . . 12 (𝑡 ∈ V → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
7876, 77mp1i 13 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
79 inss1 4120 . . . . . . . . . . . 12 (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
80 sspwb 5226 . . . . . . . . . . . . 13 (𝑡𝑋 ↔ 𝒫 𝑡 ⊆ 𝒫 𝑋)
8166, 80sylib 219 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
8279, 81syl5ss 3895 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
83 fvex 6543 . . . . . . . . . . . . 13 (mrCls‘𝐶) ∈ V
84 imaexg 7467 . . . . . . . . . . . . 13 ((mrCls‘𝐶) ∈ V → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
8583, 84ax-mp 5 . . . . . . . . . . . 12 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V
8685a1i 11 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
8772, 75, 78, 82, 86ipodrsima 17592 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
8887adantlr 711 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
89 imassrn 5809 . . . . . . . . . . . . . 14 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
9070frnd 6381 . . . . . . . . . . . . . 14 (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
9189, 90syl5ss 3895 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9291adantr 481 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9385elpw 4453 . . . . . . . . . . . 12 (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9492, 93sylibr 235 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
9594adantlr 711 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
96 simplr 765 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
97 fveq2 6530 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
9897eleq1d 2865 . . . . . . . . . . . 12 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
99 unieq 4747 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → 𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
100 id 22 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → 𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
10199, 100eleq12d 2875 . . . . . . . . . . . 12 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ( 𝑠𝑠 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
10298, 101imbi12d 346 . . . . . . . . . . 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)))))
103102rspcva 3552 . . . . . . . . . 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))))
10495, 96, 103syl2anc 584 . . . . . . . . 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))))
10588, 104mpd 15 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
10669, 105eqeltrd 2881 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
107 fvelimab 6597 . . . . . . . . 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))
10872, 82, 107syl2anc 584 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
109108adantlr 711 . . . . . . 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))
110106, 109mpbid 233 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
111 eqcom 2800 . . . . . . 7 (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
112111rexbii 3209 . . . . . 6 (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
113110, 112sylibr 235 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11410mrefg2 38739 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
115114ad2antrr 722 . . . . 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)))
116113, 115mpbird 258 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
117116ralrimiva 3147 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11810isnacs 38736 . . 3 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
11962, 117, 118sylanbrc 583 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
12054, 119impbii 210 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 207   ∧ wa 396   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∃wrex 3104  Vcvv 3432   ∩ cin 3853   ⊆ wss 3854  𝒫 cpw 4447  ∪ cuni 4739  ran crn 5436   “ cima 5438   Fn wfn 6212  ‘cfv 6217  Fincfn 8347  Moorecmre 16670  mrClscmrc 16671  ACScacs 16673  Dirsetcdrs 17354  toInccipo 17578  NoeACScnacs 38734 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-z 11819  df-dec 11937  df-uz 12083  df-fz 12732  df-struct 16302  df-ndx 16303  df-slot 16304  df-base 16306  df-tset 16401  df-ple 16402  df-ocomp 16403  df-mre 16674  df-mrc 16675  df-acs 16677  df-proset 17355  df-drs 17356  df-poset 17373  df-ipo 17579  df-nacs 38735 This theorem is referenced by:  nacsfix  38744
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