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Theorem isnacs3 42697
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 42696 . . . 4 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
21acsmred 17700 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3 simpll 767 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
41ad2antrr 726 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5 elpwi 4611 . . . . . . . . . 10 (𝑠 ∈ 𝒫 𝐶𝑠𝐶)
65ad2antlr 727 . . . . . . . . 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 18600 . . . . . . . . 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 2734 . . . . . . . . 9 (mrCls‘𝐶) = (mrCls‘𝐶)
1110nacsfg 42692 . . . . . . . 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 42694 . . . . . . . . 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 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)))
1612, 15mpbid 232 . . . . . 6 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
17 elfpw 9391 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑔 𝑠𝑔 ∈ Fin))
18 fissuni 9394 . . . . . . . . 9 ((𝑔 𝑠𝑔 ∈ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
1917, 18sylbi 217 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
20 elfpw 9391 . . . . . . . . . . . 12 ( ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑠 ∈ Fin))
21 ipodrsfi 18596 . . . . . . . . . . . . 13 (((toInc‘𝑠) ∈ Dirset ∧ 𝑠 ∈ Fin) → ∃𝑖𝑠 𝑖)
22213expb 1119 . . . . . . . . . . . 12 (((toInc‘𝑠) ∈ Dirset ∧ (𝑠 ∈ Fin)) → ∃𝑖𝑠 𝑖)
2320, 22sylan2b 594 . . . . . . . . . . 11 (((toInc‘𝑠) ∈ Dirset ∧ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖𝑠 𝑖)
24 sstr 4003 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28 simprr 773 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑔𝑖)
295ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑠𝐶)
30 simprl 771 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝑠)
3129, 30sseldd 3995 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝐶)
3210mrcsscl 17664 . . . . . . . . . . . . . . . . . . . . 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 4033 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑖)
36 simplrl 777 . . . . . . . . . . . . . . . . . . 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 4012 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = 𝑖)
4039, 36eqeltrd 2838 . . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . 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 3150 . . . . . . . 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 3150 . . . . . 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 3143 . . 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 3993 . . . . . . 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 3164 . . . . 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 18607 . . . 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 17657 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
6463adantlr 715 . . . . . . . . 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 17637 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝑡𝑋)
6766adantlr 715 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡𝑋)
6865, 10, 67acsficld 18608 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
6964, 68eqtr3d 2776 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
7010mrcf 17653 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋𝐶)
7170ffnd 6737 . . . . . . . . . . . 12 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7271adantr 480 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7310mrcss 17660 . . . . . . . . . . . . 13 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
74733expb 1119 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
7574adantlr 715 . . . . . . . . . . 11 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
76 vex 3481 . . . . . . . . . . . 12 𝑡 ∈ V
77 fpwipodrs 18597 . . . . . . . . . . . 12 (𝑡 ∈ V → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
7876, 77mp1i 13 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
79 inss1 4244 . . . . . . . . . . . 12 (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
8066sspwd 4617 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
8179, 80sstrid 4006 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
82 fvex 6919 . . . . . . . . . . . . 13 (mrCls‘𝐶) ∈ V
83 imaexg 7935 . . . . . . . . . . . . 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 18598 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
8786adantlr 715 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
88 imassrn 6090 . . . . . . . . . . . . . 14 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
8970frnd 6744 . . . . . . . . . . . . . 14 (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
9088, 89sstrid 4006 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9190adantr 480 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9284elpw 4608 . . . . . . . . . . . 12 (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9391, 92sylibr 234 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
9493adantlr 715 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
95 simplr 769 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
96 fveq2 6906 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
9796eleq1d 2823 . . . . . . . . . . . 12 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
98 unieq 4922 . . . . . . . . . . . . 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 2832 . . . . . . . . . . . 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 3619 . . . . . . . . . 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 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))))
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 2838 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
106 fvelimab 6980 . . . . . . . . 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 584 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
108107adantlr 715 . . . . . . 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 232 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
110 eqcom 2741 . . . . . . 7 (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
111110rexbii 3091 . . . . . 6 (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
112109, 111sylibr 234 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11310mrefg2 42694 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
114113ad2antrr 726 . . . . 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 3143 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11710isnacs 42691 . . 3 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
11862, 116, 117sylanbrc 583 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
11954, 118impbii 209 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 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cin 3961  wss 3962  𝒫 cpw 4604   cuni 4911  ran crn 5689  cima 5691   Fn wfn 6557  cfv 6562  Fincfn 8983  Moorecmre 17626  mrClscmrc 17627  ACScacs 17629  Dirsetcdrs 18350  toInccipo 18584  NoeACScnacs 42689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-tset 17316  df-ple 17317  df-ocomp 17318  df-mre 17630  df-mrc 17631  df-acs 17633  df-proset 18351  df-drs 18352  df-poset 18370  df-ipo 18585  df-nacs 42690
This theorem is referenced by:  nacsfix  42699
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