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Theorem mreacs 17701
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 6906 . . 3 (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
2 pweq 4614 . . . 4 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
32fveq2d 6910 . . 3 (𝑥 = 𝑋 → (Moore‘𝒫 𝑥) = (Moore‘𝒫 𝑋))
41, 3eleq12d 2835 . 2 (𝑥 = 𝑋 → ((ACS‘𝑥) ∈ (Moore‘𝒫 𝑥) ↔ (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)))
5 acsmre 17695 . . . . . . 7 (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ (Moore‘𝑥))
6 mresspw 17635 . . . . . . . 8 (𝑎 ∈ (Moore‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
75, 6syl 17 . . . . . . 7 (𝑎 ∈ (ACS‘𝑥) → 𝑎 ⊆ 𝒫 𝑥)
85, 7elpwd 4606 . . . . . 6 (𝑎 ∈ (ACS‘𝑥) → 𝑎 ∈ 𝒫 𝒫 𝑥)
98ssriv 3987 . . . . 5 (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥
109a1i 11 . . . 4 (⊤ → (ACS‘𝑥) ⊆ 𝒫 𝒫 𝑥)
11 vex 3484 . . . . . . . 8 𝑥 ∈ V
12 mremre 17647 . . . . . . . 8 (𝑥 ∈ V → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
1311, 12mp1i 13 . . . . . . 7 (𝑎 ⊆ (ACS‘𝑥) → (Moore‘𝑥) ∈ (Moore‘𝒫 𝑥))
145ssriv 3987 . . . . . . . 8 (ACS‘𝑥) ⊆ (Moore‘𝑥)
15 sstr 3992 . . . . . . . 8 ((𝑎 ⊆ (ACS‘𝑥) ∧ (ACS‘𝑥) ⊆ (Moore‘𝑥)) → 𝑎 ⊆ (Moore‘𝑥))
1614, 15mpan2 691 . . . . . . 7 (𝑎 ⊆ (ACS‘𝑥) → 𝑎 ⊆ (Moore‘𝑥))
17 mrerintcl 17640 . . . . . . 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 3978 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → 𝑑 ∈ (ACS‘𝑥))
2019acsmred 17699 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → 𝑑 ∈ (Moore‘𝑥))
21 eqid 2737 . . . . . . . . . . . . . . 15 (mrCls‘𝑑) = (mrCls‘𝑑)
2220, 21mrcssvd 17666 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2322ralrimiva 3146 . . . . . . . . . . . . 13 (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2423adantr 480 . . . . . . . . . . . 12 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
25 iunss 5045 . . . . . . . . . . . 12 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥 ↔ ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2624, 25sylibr 234 . . . . . . . . . . 11 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2711elpw2 5334 . . . . . . . . . . 11 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ⊆ 𝑥)
2826, 27sylibr 234 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑐 ∈ 𝒫 𝑥) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) ∈ 𝒫 𝑥)
2928fmpttd 7135 . . . . . . . . 9 (𝑎 ⊆ (ACS‘𝑥) → (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)):𝒫 𝑥⟶𝒫 𝑥)
30 fssxp 6763 . . . . . . . . 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 5377 . . . . . . . . 9 𝒫 𝑥 ∈ V
3332, 32xpex 7773 . . . . . . . 8 (𝒫 𝑥 × 𝒫 𝑥) ∈ V
34 ssexg 5323 . . . . . . . 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 715 . . . . . . . . . . . . 13 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑𝑎) → 𝑑 ∈ (ACS‘𝑥))
37 elpwi 4607 . . . . . . . . . . . . . 14 (𝑏 ∈ 𝒫 𝑥𝑏𝑥)
3837ad2antlr 727 . . . . . . . . . . . . 13 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑑𝑎) → 𝑏𝑥)
3921acsfiel2 17698 . . . . . . . . . . . . 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 3176 . . . . . . . . . . 11 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑𝑎 𝑏𝑑 ↔ ∀𝑑𝑎𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
42 iunss 5045 . . . . . . . . . . . . 13 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
4342ralbii 3093 . . . . . . . . . . . 12 (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
44 ralcom 3289 . . . . . . . . . . . 12 (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin)∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑𝑎𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
4543, 44bitri 275 . . . . . . . . . . 11 (∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏 ↔ ∀𝑑𝑎𝑒 ∈ (𝒫 𝑏 ∩ Fin)((mrCls‘𝑑)‘𝑒) ⊆ 𝑏)
4641, 45bitr4di 289 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (∀𝑑𝑎 𝑏𝑑 ↔ ∀𝑒 ∈ (𝒫 𝑏 ∩ Fin) 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑏))
47 elrint2 4990 . . . . . . . . . . 11 (𝑏 ∈ 𝒫 𝑥 → (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ∀𝑑𝑎 𝑏𝑑))
4847adantl 481 . . . . . . . . . 10 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ∀𝑑𝑎 𝑏𝑑))
49 funmpt 6604 . . . . . . . . . . . . 13 Fun (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))
50 funiunfv 7268 . . . . . . . . . . . . 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 4012 . . . . . . . . . . 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 5045 . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 15 (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))
55 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑒 → ((mrCls‘𝑑)‘𝑐) = ((mrCls‘𝑑)‘𝑒))
5655iuneq2d 5022 . . . . . . . . . . . . . . 15 (𝑐 = 𝑒 𝑑𝑎 ((mrCls‘𝑑)‘𝑐) = 𝑑𝑎 ((mrCls‘𝑑)‘𝑒))
57 inss1 4237 . . . . . . . . . . . . . . . . 17 (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑏
5837sspwd 4613 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ 𝒫 𝑥 → 𝒫 𝑏 ⊆ 𝒫 𝑥)
5958adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → 𝒫 𝑏 ⊆ 𝒫 𝑥)
6057, 59sstrid 3995 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝒫 𝑏 ∩ Fin) ⊆ 𝒫 𝑥)
6160sselda 3983 . . . . . . . . . . . . . . 15 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑒 ∈ 𝒫 𝑥)
6220, 21mrcssvd 17666 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑑𝑎) → ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
6362ralrimiva 3146 . . . . . . . . . . . . . . . . . 18 (𝑎 ⊆ (ACS‘𝑥) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
6463ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
65 iunss 5045 . . . . . . . . . . . . . . . . 17 ( 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥 ↔ ∀𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
6664, 65sylibr 234 . . . . . . . . . . . . . . . 16 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥)
67 ssexg 5323 . . . . . . . . . . . . . . . 16 (( 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ⊆ 𝑥𝑥 ∈ V) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
6866, 11, 67sylancl 586 . . . . . . . . . . . . . . 15 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → 𝑑𝑎 ((mrCls‘𝑑)‘𝑒) ∈ V)
6954, 56, 61, 68fvmptd3 7039 . . . . . . . . . . . . . 14 (((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) ∧ 𝑒 ∈ (𝒫 𝑏 ∩ Fin)) → ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐))‘𝑒) = 𝑑𝑎 ((mrCls‘𝑑)‘𝑒))
7069sseq1d 4015 . . . . . . . . . . . . 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))
7170ralbidva 3176 . . . . . . . . . . . 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))
7253, 71bitrid 283 . . . . . . . . . . 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))
7352, 72bitr3id 285 . . . . . . . . . 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))
7446, 48, 733bitr4d 311 . . . . . . . . 9 ((𝑎 ⊆ (ACS‘𝑥) ∧ 𝑏 ∈ 𝒫 𝑥) → (𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
7574ralrimiva 3146 . . . . . . . 8 (𝑎 ⊆ (ACS‘𝑥) → ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏))
7629, 75jca 511 . . . . . . 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)))
77 feq1 6716 . . . . . . . 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))
78 imaeq1 6073 . . . . . . . . . . . 12 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
7978unieqd 4920 . . . . . . . . . . 11 (𝑓 = (𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) → (𝑓 “ (𝒫 𝑏 ∩ Fin)) = ((𝑐 ∈ 𝒫 𝑥 𝑑𝑎 ((mrCls‘𝑑)‘𝑐)) “ (𝒫 𝑏 ∩ Fin)))
8079sseq1d 4015 . . . . . . . . . 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))
8180bibi2d 342 . . . . . . . . 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)))
8281ralbidv 3178 . . . . . . . 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)))
8377, 82anbi12d 632 . . . . . . 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))))
8435, 76, 83spcedv 3598 . . . . . 6 (𝑎 ⊆ (ACS‘𝑥) → ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑏 ∈ 𝒫 𝑥(𝑏 ∈ (𝒫 𝑥 𝑎) ↔ (𝑓 “ (𝒫 𝑏 ∩ Fin)) ⊆ 𝑏)))
85 isacs 17694 . . . . . 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))))
8618, 84, 85sylanbrc 583 . . . . 5 (𝑎 ⊆ (ACS‘𝑥) → (𝒫 𝑥 𝑎) ∈ (ACS‘𝑥))
8786adantl 481 . . . 4 ((⊤ ∧ 𝑎 ⊆ (ACS‘𝑥)) → (𝒫 𝑥 𝑎) ∈ (ACS‘𝑥))
8810, 87ismred2 17646 . . 3 (⊤ → (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥))
8988mptru 1547 . 2 (ACS‘𝑥) ∈ (Moore‘𝒫 𝑥)
904, 89vtoclg 3554 1 (𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wtru 1541  wex 1779  wcel 2108  wral 3061  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   cint 4946   ciun 4991  cmpt 5225   × cxp 5683  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  Fincfn 8985  Moorecmre 17625  mrClscmrc 17626  ACScacs 17628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630  df-acs 17632
This theorem is referenced by:  acsfn1  17704  acsfn1c  17705  acsfn2  17706  submacs  18840  subgacs  19179  nsgacs  19180  acsfn1p  20800  subrgacs  20801  sdrgacs  20802  lssacs  20965
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