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Theorem isacs1i 17700
Description: A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
isacs1i ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))
Distinct variable groups:   𝐹,𝑠   𝑋,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem isacs1i
Dummy variables 𝑎 𝑡 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 4080 . . . 4 {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ⊆ 𝒫 𝑋
21a1i 11 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ⊆ 𝒫 𝑋)
3 pweq 4614 . . . . . . . 8 (𝑠 = (𝑋 𝑡) → 𝒫 𝑠 = 𝒫 (𝑋 𝑡))
43ineq1d 4219 . . . . . . 7 (𝑠 = (𝑋 𝑡) → (𝒫 𝑠 ∩ Fin) = (𝒫 (𝑋 𝑡) ∩ Fin))
54imaeq2d 6078 . . . . . 6 (𝑠 = (𝑋 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)))
65unieqd 4920 . . . . 5 (𝑠 = (𝑋 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)))
7 id 22 . . . . 5 (𝑠 = (𝑋 𝑡) → 𝑠 = (𝑋 𝑡))
86, 7sseq12d 4017 . . . 4 (𝑠 = (𝑋 𝑡) → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡)))
9 inss1 4237 . . . . . 6 (𝑋 𝑡) ⊆ 𝑋
10 elpw2g 5333 . . . . . 6 (𝑋𝑉 → ((𝑋 𝑡) ∈ 𝒫 𝑋 ↔ (𝑋 𝑡) ⊆ 𝑋))
119, 10mpbiri 258 . . . . 5 (𝑋𝑉 → (𝑋 𝑡) ∈ 𝒫 𝑋)
1211ad2antrr 726 . . . 4 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 𝑡) ∈ 𝒫 𝑋)
13 imassrn 6089 . . . . . . . . 9 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ ran 𝐹
14 frn 6743 . . . . . . . . . 10 (𝐹:𝒫 𝑋⟶𝒫 𝑋 → ran 𝐹 ⊆ 𝒫 𝑋)
1514adantl 481 . . . . . . . . 9 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → ran 𝐹 ⊆ 𝒫 𝑋)
1613, 15sstrid 3995 . . . . . . . 8 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
1716unissd 4917 . . . . . . 7 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
18 unipw 5455 . . . . . . 7 𝒫 𝑋 = 𝑋
1917, 18sseqtrdi 4024 . . . . . 6 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑋)
2019adantr 480 . . . . 5 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑋)
21 inss2 4238 . . . . . . . . . . . . . 14 (𝑋 𝑡) ⊆ 𝑡
22 intss1 4963 . . . . . . . . . . . . . 14 (𝑎𝑡 𝑡𝑎)
2321, 22sstrid 3995 . . . . . . . . . . . . 13 (𝑎𝑡 → (𝑋 𝑡) ⊆ 𝑎)
2423adantl 481 . . . . . . . . . . . 12 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝑋 𝑡) ⊆ 𝑎)
2524sspwd 4613 . . . . . . . . . . 11 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → 𝒫 (𝑋 𝑡) ⊆ 𝒫 𝑎)
2625ssrind 4244 . . . . . . . . . 10 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin))
27 imass2 6120 . . . . . . . . . 10 ((𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
2826, 27syl 17 . . . . . . . . 9 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
2928unissd 4917 . . . . . . . 8 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
30 ssel2 3978 . . . . . . . . . 10 ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎𝑡) → 𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
31 pweq 4614 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → 𝒫 𝑠 = 𝒫 𝑎)
3231ineq1d 4219 . . . . . . . . . . . . . . 15 (𝑠 = 𝑎 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑎 ∩ Fin))
3332imaeq2d 6078 . . . . . . . . . . . . . 14 (𝑠 = 𝑎 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
3433unieqd 4920 . . . . . . . . . . . . 13 (𝑠 = 𝑎 (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
35 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑎𝑠 = 𝑎)
3634, 35sseq12d 4017 . . . . . . . . . . . 12 (𝑠 = 𝑎 → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
3736elrab 3692 . . . . . . . . . . 11 (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑎 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
3837simprbi 496 . . . . . . . . . 10 (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3930, 38syl 17 . . . . . . . . 9 ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎𝑡) → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
4039adantll 714 . . . . . . . 8 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
4129, 40sstrd 3994 . . . . . . 7 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
4241ralrimiva 3146 . . . . . 6 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∀𝑎𝑡 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
43 ssint 4964 . . . . . 6 ( (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑡 ↔ ∀𝑎𝑡 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
4442, 43sylibr 234 . . . . 5 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑡)
4520, 44ssind 4241 . . . 4 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡))
468, 12, 45elrabd 3694 . . 3 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 𝑡) ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
472, 46ismred2 17646 . 2 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋))
48 fssxp 6763 . . . 4 (𝐹:𝒫 𝑋⟶𝒫 𝑋𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋))
49 pwexg 5378 . . . . 5 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
5049, 49xpexd 7771 . . . 4 (𝑋𝑉 → (𝒫 𝑋 × 𝒫 𝑋) ∈ V)
51 ssexg 5323 . . . 4 ((𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋) ∧ (𝒫 𝑋 × 𝒫 𝑋) ∈ V) → 𝐹 ∈ V)
5248, 50, 51syl2anr 597 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹 ∈ V)
53 simpr 484 . . . 4 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
54 pweq 4614 . . . . . . . . . 10 (𝑠 = 𝑡 → 𝒫 𝑠 = 𝒫 𝑡)
5554ineq1d 4219 . . . . . . . . 9 (𝑠 = 𝑡 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑡 ∩ Fin))
5655imaeq2d 6078 . . . . . . . 8 (𝑠 = 𝑡 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
5756unieqd 4920 . . . . . . 7 (𝑠 = 𝑡 (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
58 id 22 . . . . . . 7 (𝑠 = 𝑡𝑠 = 𝑡)
5957, 58sseq12d 4017 . . . . . 6 (𝑠 = 𝑡 → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6059elrab3 3693 . . . . 5 (𝑡 ∈ 𝒫 𝑋 → (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6160rgen 3063 . . . 4 𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)
6253, 61jctir 520 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
63 feq1 6716 . . . 4 (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋𝐹:𝒫 𝑋⟶𝒫 𝑋))
64 imaeq1 6073 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
6564unieqd 4920 . . . . . . 7 (𝑓 = 𝐹 (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
6665sseq1d 4015 . . . . . 6 (𝑓 = 𝐹 → ( (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6766bibi2d 342 . . . . 5 (𝑓 = 𝐹 → ((𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
6867ralbidv 3178 . . . 4 (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
6963, 68anbi12d 632 . . 3 (𝑓 = 𝐹 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
7052, 62, 69spcedv 3598 . 2 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
71 isacs 17694 . 2 ({𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋) ↔ ({𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
7247, 70, 71sylanbrc 583 1 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   cint 4946   × cxp 5683  ran crn 5686  cima 5688  wf 6557  cfv 6561  Fincfn 8985  Moorecmre 17625  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-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-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-acs 17632
This theorem is referenced by:  acsfn  17702
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