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Theorem isnacs3 36774
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 36773 . . . 4 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
21acsmred 16241 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
3 simpll 789 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (NoeACS‘𝑋))
41ad2antrr 761 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝐶 ∈ (ACS‘𝑋))
5 elpwi 4142 . . . . . . . . . 10 (𝑠 ∈ 𝒫 𝐶𝑠𝐶)
65ad2antlr 762 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
7 simpr 477 . . . . . . . . 9 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → (toInc‘𝑠) ∈ Dirset)
8 acsdrsel 17091 . . . . . . . . 9 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑠𝐶 ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
94, 6, 7, 8syl3anc 1323 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → 𝑠𝐶)
10 eqid 2621 . . . . . . . . 9 (mrCls‘𝐶) = (mrCls‘𝐶)
1110nacsfg 36769 . . . . . . . 8 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
123, 9, 11syl2anc 692 . . . . . . 7 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
1310mrefg2 36771 . . . . . . . . 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 761 . . . . . . 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 222 . . . . . 6 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (toInc‘𝑠) ∈ Dirset) → ∃𝑔 ∈ (𝒫 𝑠 ∩ Fin) 𝑠 = ((mrCls‘𝐶)‘𝑔))
17 elfpw 8215 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑔 𝑠𝑔 ∈ Fin))
18 fissuni 8218 . . . . . . . . 9 ((𝑔 𝑠𝑔 ∈ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
1917, 18sylbi 207 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑠 ∩ Fin) → ∃ ∈ (𝒫 𝑠 ∩ Fin)𝑔 )
20 elfpw 8215 . . . . . . . . . . . 12 ( ∈ (𝒫 𝑠 ∩ Fin) ↔ (𝑠 ∈ Fin))
21 ipodrsfi 17087 . . . . . . . . . . . . 13 (((toInc‘𝑠) ∈ Dirset ∧ 𝑠 ∈ Fin) → ∃𝑖𝑠 𝑖)
22213expb 1263 . . . . . . . . . . . 12 (((toInc‘𝑠) ∈ Dirset ∧ (𝑠 ∈ Fin)) → ∃𝑖𝑠 𝑖)
2320, 22sylan2b 492 . . . . . . . . . . 11 (((toInc‘𝑠) ∈ Dirset ∧ ∈ (𝒫 𝑠 ∩ Fin)) → ∃𝑖𝑠 𝑖)
24 sstr 3592 . . . . . . . . . . . . . . 15 ((𝑔 𝑖) → 𝑔𝑖)
2524ancoms 469 . . . . . . . . . . . . . 14 (( 𝑖𝑔 ) → 𝑔𝑖)
26 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = ((mrCls‘𝐶)‘𝑔))
272ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝐶 ∈ (Moore‘𝑋))
28 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑔𝑖)
295ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑠𝐶)
30 simprl 793 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝑠)
3129, 30sseldd 3585 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → 𝑖𝐶)
3210mrcsscl 16204 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑖𝑖𝐶) → ((mrCls‘𝐶)‘𝑔) ⊆ 𝑖)
3327, 28, 31, 32syl3anc 1323 . . . . . . . . . . . . . . . . . . . 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 3620 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑖)
36 simplrl 799 . . . . . . . . . . . . . . . . . . 19 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖𝑠)
37 elssuni 4435 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑠𝑖 𝑠)
3836, 37syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑖 𝑠)
3935, 38eqssd 3601 . . . . . . . . . . . . . . . . 17 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠 = 𝑖)
4039, 36eqeltrd 2698 . . . . . . . . . . . . . . . 16 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) ∧ 𝑠 = ((mrCls‘𝐶)‘𝑔)) → 𝑠𝑠)
4140ex 450 . . . . . . . . . . . . . . 15 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ (𝑖𝑠𝑔𝑖)) → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))
4241expr 642 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . 12 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) ∧ 𝑖𝑠) → ( 𝑖 → (𝑔 → ( 𝑠 = ((mrCls‘𝐶)‘𝑔) → 𝑠𝑠))))
4544rexlimdva 3024 . . . . . . . . . . 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 3023 . . . . . . . 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 3023 . . . . . 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 450 . . . 4 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑠 ∈ 𝒫 𝐶) → ((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
5352ralrimiva 2960 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
542, 53jca 554 . 2 (𝐶 ∈ (NoeACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)))
55 simpl 473 . . . 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 3583 . . . . . . 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 2956 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
6059imp 445 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶))
61 isacs3 17098 . . . 4 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))
6255, 60, 61sylanbrc 697 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (ACS‘𝑋))
6310mrcid 16197 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = 𝑡)
6463adantlr 750 . . . . . . . . 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 16177 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝑡𝑋)
6766adantlr 750 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡𝑋)
6865, 10, 67acsficld 17099 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶)‘𝑡) = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
6964, 68eqtr3d 2657 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
7010mrcf 16193 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶):𝒫 𝑋𝐶)
71 ffn 6004 . . . . . . . . . . . . 13 ((mrCls‘𝐶):𝒫 𝑋𝐶 → (mrCls‘𝐶) Fn 𝒫 𝑋)
7270, 71syl 17 . . . . . . . . . . . 12 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7372adantr 481 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (mrCls‘𝐶) Fn 𝒫 𝑋)
7410mrcss 16200 . . . . . . . . . . . . 13 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑔𝑋) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
75743expb 1263 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
7675adantlr 750 . . . . . . . . . . 11 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) ∧ (𝑔𝑋)) → ((mrCls‘𝐶)‘𝑔) ⊆ ((mrCls‘𝐶)‘))
77 vex 3189 . . . . . . . . . . . 12 𝑡 ∈ V
78 fpwipodrs 17088 . . . . . . . . . . . 12 (𝑡 ∈ V → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
7977, 78mp1i 13 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘(𝒫 𝑡 ∩ Fin)) ∈ Dirset)
80 inss1 3813 . . . . . . . . . . . 12 (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑡
81 sspwb 4880 . . . . . . . . . . . . 13 (𝑡𝑋 ↔ 𝒫 𝑡 ⊆ 𝒫 𝑋)
8266, 81sylib 208 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → 𝒫 𝑡 ⊆ 𝒫 𝑋)
8380, 82syl5ss 3595 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝒫 𝑡 ∩ Fin) ⊆ 𝒫 𝑋)
84 fvex 6160 . . . . . . . . . . . . 13 (mrCls‘𝐶) ∈ V
85 imaexg 7053 . . . . . . . . . . . . 13 ((mrCls‘𝐶) ∈ V → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
8684, 85ax-mp 5 . . . . . . . . . . . 12 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V
8786a1i 11 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ V)
8873, 76, 79, 83, 87ipodrsima 17089 . . . . . . . . . 10 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
8988adantlr 750 . . . . . . . . 9 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset)
90 imassrn 5438 . . . . . . . . . . . . . 14 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ ran (mrCls‘𝐶)
91 frn 6012 . . . . . . . . . . . . . . 15 ((mrCls‘𝐶):𝒫 𝑋𝐶 → ran (mrCls‘𝐶) ⊆ 𝐶)
9270, 91syl 17 . . . . . . . . . . . . . 14 (𝐶 ∈ (Moore‘𝑋) → ran (mrCls‘𝐶) ⊆ 𝐶)
9390, 92syl5ss 3595 . . . . . . . . . . . . 13 (𝐶 ∈ (Moore‘𝑋) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9493adantr 481 . . . . . . . . . . . 12 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9586elpw 4138 . . . . . . . . . . . 12 (((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶 ↔ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝐶)
9694, 95sylibr 224 . . . . . . . . . . 11 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
9796adantlr 750 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ 𝒫 𝐶)
98 simplr 791 . . . . . . . . . 10 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠))
99 fveq2 6150 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → (toInc‘𝑠) = (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
10099eleq1d 2683 . . . . . . . . . . . 12 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))) ∈ Dirset))
101 unieq 4412 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → 𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
102 id 22 . . . . . . . . . . . . 13 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → 𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
103101, 102eleq12d 2692 . . . . . . . . . . . 12 (𝑠 = ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) → ( 𝑠𝑠 ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin))))
104100, 103imbi12d 334 . . . . . . . . . . 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)))))
105104rspcva 3293 . . . . . . . . . 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))))
10697, 98, 105syl2anc 692 . . . . . . . . 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))))
10789, 106mpd 15 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
10869, 107eqeltrd 2698 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → 𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)))
109 fvelimab 6212 . . . . . . . . 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))
11073, 83, 109syl2anc 692 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡𝐶) → (𝑡 ∈ ((mrCls‘𝐶) “ (𝒫 𝑡 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡))
111110adantlr 750 . . . . . . 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))
112108, 111mpbid 222 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
113 eqcom 2628 . . . . . . 7 (𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ((mrCls‘𝐶)‘𝑔) = 𝑡)
114113rexbii 3034 . . . . . 6 (∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)((mrCls‘𝐶)‘𝑔) = 𝑡)
115112, 114sylibr 224 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
11610mrefg2 36771 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑡 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
117116ad2antrr 761 . . . . 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)))
118115, 117mpbird 247 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) ∧ 𝑡𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
119118ralrimiva 2960 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔))
12010isnacs 36768 . . 3 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑡𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑡 = ((mrCls‘𝐶)‘𝑔)))
12162, 119, 120sylanbrc 697 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)) → 𝐶 ∈ (NoeACS‘𝑋))
12254, 121impbii 199 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cin 3555  wss 3556  𝒫 cpw 4132   cuni 4404  ran crn 5077  cima 5079   Fn wfn 5844  wf 5845  cfv 5849  Fincfn 7902  Moorecmre 16166  mrClscmrc 16167  ACScacs 16169  Dirsetcdrs 16851  toInccipo 17075  NoeACScnacs 36766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-dec 11441  df-uz 11635  df-fz 12272  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-tset 15884  df-ple 15885  df-ocomp 15887  df-mre 16170  df-mrc 16171  df-acs 16173  df-preset 16852  df-drs 16853  df-poset 16870  df-ipo 17076  df-nacs 36767
This theorem is referenced by:  nacsfix  36776
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