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Theorem mreacs 16924
Description: Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mreacs (𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

Proof of Theorem mreacs
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6669 . . 3 (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
2 pweq 4545 . . . 4 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
32fveq2d 6673 . . 3 (𝑥 = 𝑋 → (Moore‘𝒫 𝑥) = (Moore‘𝒫 𝑋))
41, 3eleq12d 2912 . 2 (𝑥 = 𝑋 → ((ACS‘𝑥) ∈ (Moore‘𝒫 𝑥) ↔ (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)))
5 acsmre 16918 . . . . . . 7 (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ (Moore‘𝑥))
6 mresspw 16858 . . . . . . . 8 (𝑎 ∈ (Moore‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
75, 6syl 17 . . . . . . 7 (𝑎 ∈ (ACS‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
85, 7elpwd 4553 . . . . . 6 (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ 𝒫 𝒫 𝑥)
98ssriv 3975 . . . . 5 (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥
109a1i 11 . . . 4 (⊤ → (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥)
11 vex 3503 . . . . . . . 8 𝑥 ∈ V
12 mremre 16870 . . . . . . . 8 (𝑥 ∈ V → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
1311, 12mp1i 13 . . . . . . 7 (𝑎 ⊆ (ACS‘𝑥) → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
145ssriv 3975 . . . . . . . 8 (ACS‘𝑥) ⊆ (Moore‘𝑥)
15 sstr 3979 . . . . . . . 8 ((𝑎 ⊆ (ACS‘𝑥) ∧ (ACS‘𝑥) ⊆ (Moore‘𝑥)) → 𝑎 ⊆ (Moore‘𝑥))
1614, 15mpan2 687 . . . . . . 7 (𝑎 ⊆ (ACS‘𝑥) → 𝑎 ⊆ (Moore‘𝑥))
17 mrerintcl 16863 . . . . . . 7 (((Moore‘𝑥) ∈ (Moore‘𝒫 𝑥) ∧ 𝑎 ⊆ (Moore‘𝑥)) → (𝒫 𝑥 𝑎) ∈ (Moore‘𝑥))
1813, 16, 17syl2anc 584 . . . . . 6 (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 𝑎) ∈ (Moore‘𝑥))
19 ssel2 3966 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → 𝑑 ∈ (ACS‘𝑥))
2019acsmred 16922 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → 𝑑 ∈ (Moore‘𝑥))
21 eqid 2826 . . . . . . . . . . . . . . 15 (mrCls‘𝑑) = (mrCls‘𝑑)
2220, 21mrcssvd 16889 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2322ralrimiva 3187 . . . . . . . . . . . . 13 (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2423adantr 481 . . . . . . . . . . . 12 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
25 iunss 4966 . . . . . . . . . . . 12 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥 ↔ ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2624, 25sylibr 235 . . . . . . . . . . 11 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2711elpw2 5245 . . . . . . . . . . 11 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2826, 27sylibr 235 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥)
2928fmpttd 6877 . . . . . . . . 9 (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥)
30 fssxp 6533 . . . . . . . . 9 ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 → (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥))
3129, 30syl 17 . . . . . . . 8 (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥))
32 vpwex 5275 . . . . . . . . 9 𝒫 𝑥 ∈ V
3332, 32xpex 7469 . . . . . . . 8 (𝒫 𝑥 × 𝒫 𝑥) ∈ V
34 ssexg 5224 . . . . . . . 8 (((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) ⊆ (𝒫 𝑥 × 𝒫 𝑥) ∧ (𝒫 𝑥 × 𝒫 𝑥) ∈ V) → (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) ∈ V)
3531, 33, 34sylancl 586 . . . . . . 7 (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) ∈ V)
3619adantlr 711 . . . . . . . . . . . . 13 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑𝑎) → 𝑑 ∈ (ACS‘𝑥))
37 elpwi 4554 . . . . . . . . . . . . . 14 (𝑏 ∈ 𝒫 𝑥𝑏𝑥)
3837ad2antlr 723 . . . . . . . . . . . . 13 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑𝑎) → 𝑏𝑥)
3921acsfiel2 16921 . . . . . . . . . . . . 13 ((𝑑 ∈ (ACS‘𝑥) ∧ 𝑏𝑥) → (𝑏𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
4036, 38, 39syl2anc 584 . . . . . . . . . . . 12 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑𝑎) → (𝑏𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
4140ralbidva 3201 . . . . . . . . . . 11 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑𝑎 𝑏𝑑 ↔ ∀𝑑𝑎𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
42 iunss 4966 . . . . . . . . . . . . 13 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
4342ralbii 3170 . . . . . . . . . . . 12 (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
44 ralcom 3359 . . . . . . . . . . . 12 (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑𝑎𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
4543, 44bitri 276 . . . . . . . . . . 11 (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑𝑎𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
4641, 45syl6bbr 290 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑𝑎 𝑏𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
47 elrint2 4916 . . . . . . . . . . 11 (𝑏 ∈ 𝒫 𝑥 → (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ∀𝑑𝑎 𝑏𝑑))
4847adantl 482 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ∀𝑑𝑎 𝑏𝑑))
49 funmpt 6392 . . . . . . . . . . . . 13 Fun (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))
50 funiunfv 7003 . . . . . . . . . . . . 13 (Fun (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
5149, 50ax-mp 5 . . . . . . . . . . . 12 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin))
5251sseq1i 3999 . . . . . . . . . . 11 ( 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)
53 iunss 4966 . . . . . . . . . . . 12 ( 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏)
54 eqid 2826 . . . . . . . . . . . . . . 15 (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))
55 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑒 → ((mrCls‘𝑑)‘𝑐) = ((mrCls‘𝑑)‘𝑒))
5655iuneq2d 4945 . . . . . . . . . . . . . . 15 (𝑐 = 𝑒 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) = 𝑑𝑎 ((mrCls‘𝑑)‘𝑒))
57 inss1 4209 . . . . . . . . . . . . . . . . 17 (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑏
58 sspwb 5338 . . . . . . . . . . . . . . . . . . 19 (𝑏𝑥 ↔ 𝒫 𝑏 ⊆ 𝒫 𝑥)
5937, 58sylib 219 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ 𝒫 𝑥 → 𝒫 𝑏 ⊆ 𝒫 𝑥)
6059adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → 𝒫 𝑏 ⊆ 𝒫 𝑥)
6157, 60sstrid 3982 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑥)
6261sselda 3971 . . . . . . . . . . . . . . 15 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑒 ∈ 𝒫 𝑥)
6320, 21mrcssvd 16889 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
6463ralrimiva 3187 . . . . . . . . . . . . . . . . . 18 (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
6564ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
66 iunss 4966 . . . . . . . . . . . . . . . . 17 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥 ↔ ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
6765, 66sylibr 235 . . . . . . . . . . . . . . . 16 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
68 ssexg 5224 . . . . . . . . . . . . . . . 16 (( 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥𝑥 ∈ V) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
6967, 11, 68sylancl 586 . . . . . . . . . . . . . . 15 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
7054, 56, 62, 69fvmptd3 6789 . . . . . . . . . . . . . 14 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = 𝑑𝑎 ((mrCls‘𝑑)‘𝑒))
7170sseq1d 4002 . . . . . . . . . . . . 13 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → (((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
7271ralbidva 3201 . . . . . . . . . . . 12 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
7353, 72syl5bb 284 . . . . . . . . . . 11 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → ( 𝑒 ∈ (𝒫 𝑏 ∩ Fin)((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
7452, 73syl5bbr 286 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → ( ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
7546, 48, 743bitr4d 312 . . . . . . . . 9 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
7675ralrimiva 3187 . . . . . . . 8 (𝑎 ⊆ (ACS‘𝑥) → ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
7729, 76jca 512 . . . . . . 7 (𝑎 ⊆ (ACS‘𝑥) → ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
78 feq1 6494 . . . . . . . 8 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓:𝒫 𝑥⟶𝒫 𝑥 ↔ (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥))
79 imaeq1 5923 . . . . . . . . . . . 12 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
8079unieqd 4847 . . . . . . . . . . 11 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
8180sseq1d 4002 . . . . . . . . . 10 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → ( (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏 ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
8281bibi2d 344 . . . . . . . . 9 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → ((𝑏 ∈ (𝒫 𝑥 𝑎) ↔ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏) ↔ (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
8382ralbidv 3202 . . . . . . . 8 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → (∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏) ↔ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
8478, 83anbi12d 630 . . . . . . 7 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → ((𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))))
8535, 77, 84elabd 3672 . . . . . 6 (𝑎 ⊆ (ACS‘𝑥) → ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
86 isacs 16917 . . . . . 6 ((𝒫 𝑥 𝑎) ∈ (ACS‘𝑥) ↔ ((𝒫 𝑥 𝑎) ∈ (Moore‘𝑥) ∧ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))))
8718, 85, 86sylanbrc 583 . . . . 5 (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 𝑎) ∈ (ACS‘𝑥))
8887adantl 482 . . . 4 ((⊤ ∧ 𝑎 ⊆ (ACS‘𝑥)) → (𝒫 𝑥 𝑎) ∈ (ACS‘𝑥))
8910, 88ismred2 16869 . . 3 (⊤ → (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥))
9089mptru 1537 . 2 (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥)
914, 90vtoclg 3573 1 (𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wtru 1531  wex 1773  wcel 2107  wral 3143  Vcvv 3500  cin 3939  wss 3940  𝒫 cpw 4542   cuni 4837   cint 4874   ciun 4917  cmpt 5143   × cxp 5552  cima 5557  Fun wfun 6348  wf 6350  cfv 6354  Fincfn 8503  Moorecmre 16848  mrClscmrc 16849  ACScacs 16851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-mre 16852  df-mrc 16853  df-acs 16855
This theorem is referenced by:  acsfn1  16927  acsfn1c  16928  acsfn2  16929  submacs  17986  subgacs  18258  nsgacs  18259  acsfn1p  19514  subrgacs  19515  sdrgacs  19516  lssacs  19675
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